Number Series

Start every series with the simplest test: subtract each term from the next and write the differences just below the series.

1. Same difference. If all the differences are equal, the series is an AP. Continue it by adding that difference. (For the nth term or a sum of such a series, use the Progression chapter.) Differences can be negative too: 80, 74, 68 falls by 6 each time.

2. Differences that go up or down by a fixed step. Take 7, 8, 10, 13, 17. The differences are +1, +2, +3, +4, so they form a pattern of their own. The next difference is +5, and the next term is 17 + 5 = 22. Radhika's money box fills in this way if she puts in ₹1 more each day than the day before.

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3. The difference table. When the first differences are not constant, write the differences of the differences (the second differences). If those are constant, you have the rule: carry the second difference one step further, then add back upwards.

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4. Differences from a known list. The differences may be odd numbers (+1, +3, +5, …), even numbers (+2, +4, +6, …), multiples of a number (+3, +6, +9, …), squares (+1, +4, +9, …) or a GP of their own (+1, +3, +9, +27). When the differences grow very fast, check whether they are squares or powers.

The series is solved only when your rule explains every printed term, not just the first two.

When a series grows fast, it is usually being multiplied. Divide each term by the one before it and write the ratios.

1. Same ratio. If every ratio is the same (×2, ×3, ×32, ÷3), the series is a GP. Continue it with that ratio, and for the nth term or a sum go to the Progression chapter. A falling series with ratio like 13 is the same thing seen from the other side.

2. Growing multiplier. The multiplier itself may go up by one each time: ×1, ×2, ×3, ×4, … Take 3, 3, 6, 18, 72. The ratios are 1, 2, 3, 4, so the next ratio is 5 and the next term is 72 × 5 = 360.

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3. Multiply, then add or subtract (k·a + c). The ratios are not equal, but the differences form a GP. The ratio of that GP is the multiplier k. Radhika has 4, 9, 19, 39, 79. The differences are 5, 10, 20, 40, which double, so k = 2. The leftover is c = 9 − 2 × 4 = 1. The rule is ×2 + 1, and the next term is 79 × 2 + 1 = 159.

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4. Check the rule. Once you guess "×k + c", try it on three consecutive pairs before you accept it. If the added number itself changes (+1, +2, +3, …), the series belongs to the type on mixed operations.

Some series look strange until you notice that every term sits next to a perfect square or a perfect cube. This is where the table you learnt in the foundation pays off.

1. Plain squares and cubes. 25, 36, 49, 64 are squares of consecutive numbers. 1, 8, 27, 64 are cubes of 1, 2, 3, 4. The squares may skip: squares of even numbers (4, 16, 36, 64), squares of odd numbers (1, 9, 25, 49), or squares of primes (4, 9, 25, 49, 121).

2. Square or cube ± a number. Radhika is given 10, 13, 18, 25, 34. Each term is 9 more than a square: 1 + 9, 4 + 9, 9 + 9, 16 + 9, 25 + 9. The next is 36 + 9 = 45. In the same way, 6, 13, 32, 69 are cubes plus 5: 1 + 5, 8 + 5, 27 + 5, 64 + 5.

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3. How to spot it. Take the term nearest to a perfect square or a cube and see how far away it is. If the distance is the same for every term, you have the rule. Terms like 342, 511 and 728 sit one below the cubes 343, 512 and 729, and 344, 513 and 730 sit one above them, so cubes minus or plus 1 are the first thing to try.

4. Cube differences. The differences of cubes are 7, 19, 37, 61, and their second differences are 12, 18, 24, which go up by a constant 6. If a difference table ends in a constant only at the third row, the series is built from cubes.

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5. Squares with a changing part. If the leftover from the nearest square is not constant but also follows n (like n² + n, n² − n), the series belongs to the next type, which deals with products.

Some series have terms that are built by multiplying numbers which move along with the position. The trick is to break every term into its factors.

1. Two neighbours, n(n + 1). Radhika sees 240, 272, 306, 342. These are 15 × 16, 16 × 17, 17 × 18, 18 × 19. Break each term into two numbers that are next to each other. The gaps between such terms are even numbers that go up by 2.

2. Two numbers with a fixed gap. The gap may be 2, 3, 4, … Take 5, 12, 21, 32, 45. These are 1 × 5, 2 × 6, 3 × 7, 4 × 8, 5 × 9, so the factors differ by 4. The next is 6 × 10 = 60.

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3. Three neighbours. 990 and 1320 are 9 × 10 × 11 and 10 × 11 × 12. For three neighbours the gaps grow faster, and the second differences are not constant.

4. Factorials and their relatives. n! means 1 × 2 × 3 × … × n, and in a factorial series every term is the previous one multiplied by the next whole number. Close relatives are n! ± k (for example 2, 3, 7, 25, 121 are factorials plus 1) and n × n!.

5. Spotting a product series. Try to split a middle term into two close factors. If 90 splits as 9 × 10 and 72 as 8 × 9, the pattern has shown itself. When the series is not a clean product, add or subtract a small number first (many series are product − 1), and then split.

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In these series the next term does not come from a fixed operation on one term. It comes from the prime list, from the earlier terms, or from the digits.

1. Prime numbers. Radhika's series 53, 59, 61, 67, 71 is a run of consecutive primes, so the next is 73. The primes can also be changed a little: p + 1 (4, 6, 8, 12), p − 1, 2p, 3p (6, 9, 15, 21, 33), or p² (4, 9, 25, 49). Always test the primes first when the differences look random but every term is odd.

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2. Fibonacci-type series. Each term is the sum of the two terms before it. Take 4, 6, 10, 16, 26. Here 4 + 6 = 10, 6 + 10 = 16, 10 + 16 = 26, so the next is 16 + 26 = 42. The start can be anything. A clue: the differences are the earlier terms themselves.

3. Other ways of using earlier terms. The next term may be the sum of the previous three (2, 3, 4, 9, 16, 29, with 54 next), the difference of the previous two (80, 50, 30, 20, 10, 10), or their product. If the series falls and a pair of terms gives the next by subtraction, try this one.

4. Digit-based series. The next term may be the previous term plus (or minus) the sum of its digits. Radhika has 5, 10, 11, 13, 17, 25: 5 + 5 = 10, 10 + 1 = 11, 11 + 2 = 13, 13 + 4 = 17, 17 + 8 = 25, and then 25 + 7 = 32. The differences 5, 1, 2, 4, 8 are the digit sums. Other versions use the product of digits or the reversed number.

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5. Which one first? Check the primes first if the terms are all odd or all clearly prime-related. Check the sum of two neighbours when the series grows slowly and smoothly. Digits come last, because they apply only to special cases.

When the terms go up and down, or when the differences look completely mixed up, there may be two separate series sharing the same line. The first, third, fifth … terms form one series, and the second, fourth, sixth … terms form another.

1. Splitting. Radhika is given 5, 20, 8, 25, 11, 30, 14. Pick the terms in odd places: 5, 8, 11, 14. They go up by 3. Now the even places: 20, 25, 30, which go up by 5, so the missing term in place 8 is 35. Each small series can follow its own rule: add, multiply, squares, even a difference table.

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2. The place of the question mark decides. Count its place from the start of the series. If it is in an odd place, work with the odd-place terms; if in an even place, with the even-place terms. The other set only helps you confirm the pattern.

3. Alternating steps (one series, two steps). Sometimes it is a single series with steps that repeat in turn, such as +7, −3, +7, −3. Radhika writes 20, 27, 24, 31, 28. Here every second term rises by the net change, 7 − 3 = 4: 20, 24, 28. This is not two different series. The same two steps are used again and again.

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4. How to tell which is which. Write the differences. If they alternate in size or sign (+7, −3, +7, −3), it is one series with two steps. If they are not repeating but every second difference follows its own pattern (for example +3, +10, +3, +15, +3, +20), it is two separate series woven together.

When no single operation explains a series, two operations may be at work together. There are two common shapes.

1. Two operations in turn. One step is ×2, the next is −3, the next is ×2 again, and so on. Radhika writes 6, 12, 9, 18, 15, 30. The steps are ×2, −3, ×2, −3, ×2, so the next step is −3 and the next term is 27. You can spot this by looking at the differences: they swing between big and small, or between plus and minus. Other common pairs are (+1, ×2), (÷2, +2), (×3, −2) and (+2, ×3).

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2. Steps that grow. The multiplier, the added number, or both can go up with the position. Look at 3, 4, 10, 33, 136:

• 3 × 1 + 1 = 4
• 4 × 2 + 2 = 10
• 10 × 3 + 3 = 33
• 33 × 4 + 4 = 136

The next step is ×5 + 5, giving 136 × 5 + 5 = 685. Other shapes of the same family are ×2 + 1, ×3 + 1, ×4 + 1 (multiplier grows, added number fixed), and ×2 + 1², ×2 + 2², ×2 + 3² (multiplier fixed, added number is a square).

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3. How to find the steps. Divide each term by the one before and round the answer to the nearest whole number. That gives the multiplier. Then compute the leftover: next term − multiplier × term. If the leftovers are equal, it is a simple ×k + c. If they run 1, 2, 3 or 1, 4, 9, the added number follows its own pattern.

A series of fractions looks harder than it is. A fraction is two numbers written one over the other, so a fraction series is usually two series running together: one in the numerators and one in the denominators.

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1. Numerators and denominators separately. Take 25, 48, 611, 814. The numerators 2, 4, 6, 8 go up by 2; the denominators 5, 8, 11, 14 go up by 3. So the next term is 1017. The two rows can follow completely different rules: one may add, the other may double or run through squares.

Never reduce a fraction in the series to its lowest terms before you have found the rule. In the series above, 48 is printed as it is because 4 and 8 belong to the two rows. If you write it as 12, the pattern disappears.

2. Decimals. A decimal series is a whole-number series with the decimal point shifted. Radhika gets 0.1, 0.3, 0.7, 1.5, 3.1 and multiplies everything by 10: 1, 3, 7, 15, 31. Now she sees ×2 + 1 at once, so the next whole number is 63 and the next term of her series is 6.3. Solve the whole numbers, then put the point back.

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3. Mixed numbers. In a series like 113, 214, 315, read the whole parts and the fraction parts as two rows again. If that does not work, change each mixed number into an improper fraction and look at the differences.

4. Differences that are fractions. When the denominators keep changing, subtract neighbouring terms. The differences often turn out to be simple fractions such as 12, 13, 14. For example 1, 32, 116 rises by 12 and then by 13.

Sums of fractions such as 11 × 2 + 12 × 3 + … are a different question (adding many terms), and the cancelling method for them is taught in the Simplification chapter.

In a wrong-number question all the terms are printed, but one of them has been changed. Note that the rule is already present in the other terms. You do not have to guess it from four numbers; you can read it from five or six.

Step 1. Read the rule from the terms that agree. Find the differences or the ratios. Most of them will follow one pattern, and one or two will look odd. The pattern of the majority is the rule.

Step 2. Write what each term should be. Start from a term you trust and build the series again with the rule. The printed term that differs from your value is the wrong number.

Step 3. Confirm with the next term. Your corrected value must lead to the next printed term by the same rule. If it does, you have found it.

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The two-spoilt-neighbours test. A changed term in the middle spoils two differences: the one before it and the one after it. One becomes too big and the other too small by the same amount. In 4, 9, 14, 20, 24, 29 the differences are +5, +5, +6, +4, +5. The +6 and +4 sit on the two sides of 20, and 6 + 4 = 10 = 5 + 5. So 20 is wrong and the correct term is 19.

A changed first or last term spoils only one difference, because it has a neighbour on one side only. Keep that in mind when the odd difference is at the end of the row.

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The same thinking works for ratios. In a ×3 series a changed term makes one ratio a little more than 3 and the next a little less than 3, and often neither is a whole number.

Never pick the term just because one difference looks odd. Find the term that spoils both of its neighbours, or that is at the end of the row.

This is the toughest form of the chapter, because the rule itself is not plain. The series may be ×2 + 1, two series woven together, or differences that have their own pattern, and in addition one term has been changed. The method is the same as in the last type; only the rule must be read a little more carefully.

1. Multiply-and-add series (×k + c). Find k from the ratios and c from the leftovers, using the first three or four terms. Then build the series again. A wrong term makes two steps fail: the step into it and the step out of it. The real test is the step out of it: put your corrected value there, and it must lead to the next printed term.

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For example, take 4, 9, 19, 40, 79. The steps 4 → 9 → 19 are ×2 + 1. 19 × 2 + 1 = 39, not 40, and 40 × 2 + 1 = 81, not 79. But 39 × 2 + 1 = 79, so the wrong term is 40.

2. Alternating series. If the terms jump up and down, split them into odd places and even places. Each row is a simple series, and the wrong term shows up in one of the two rows. Do not mix the rows while checking.

3. Two-level differences. When the first differences are not equal, find the second differences. A wrong term disturbs three second differences around it, while the rest stay constant. In practice it is quicker to guess the first-difference pattern from the differences far away from the trouble, rebuild, and compare.

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4. Primes, Fibonacci and factorial series. Check each term against what it should be. A wrong term in a Fibonacci series spoils the next two sums, so the term that is not the sum of the two before it, while its own value would fix both later sums, is the answer.

Change one term, rebuild the whole series with the rule, and check that every other printed term now fits. Only then mark the option.