A certain sequence of integers is constructed as follows : Consider 0 and 1 as the first two numbers. The next, i.e. the third number is constructed by their sums, i.e. 1. This process of constructing the next number by the sum of the last two constructed numbers continues. Taking these numbers 0, 1, 1 as the first, second and third numbers in the sequence, what will be the 7th and 10th numbers, respectively ?
- (a)6 and 30
- (b)7 and 33
- (c)8 and 34
- (d)10 and 39
Answer
Why
Correct — C, (c) 8 and 34.
The stem describes the FIBONACCI SEQUENCE and then, unusually helpfully, fixes its indexing. Each new term is the sum of the two immediately before it, and the stem states in terms that 0, 1 and 1 are to be treated as the first, second and third numbers. So write the sequence out with its positions attached:
position 1 2 3 4 5 6 7 8 9 10 value 0 1 1 2 3 5 8 13 21 34
Each value after the third is the sum of the two to its left: 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13, 8 + 13 = 21, 13 + 21 = 34.
The seventh number is 8 and the tenth is 34, which is option (c).
The whole difficulty of the question is INDEXING, and the stem's third sentence exists to remove it. The usual convention writes the Fibonacci sequence beginning 1, 1, 2, 3, 5 …, and on that convention the seventh term is 13 and the tenth is 55 — different answers to the same words. By declaring that 0 is the first number, the paper shifts every position by one and gets 8 and 34. A candidate who skims past that sentence and applies the remembered convention will produce a pair that is not on the page at all.
CHECK THE OPTIONS AGAINST THE SEQUENCE. This is the fastest route of all and worth naming as a technique. The numbers appearing in the sequence up to and beyond the tenth term are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. Now look at what the four options offer: 6, 30, 7, 33, 8, 34, 10, 39. Only option (c) contains numbers that are in the sequence at all — 6, 7, 10, 30, 33 and 39 are none of them Fibonacci numbers. So a candidate who has written out even the first ten terms can eliminate three options without reading them carefully, whatever doubts remain about where to start counting.
The 'th' of '7th' and '10th' is printed as a superscript in the booklet, and the stem is set as an opening line followed by an unindented paragraph, which is where the break in the transcribed text comes from. 'i.e.' appears twice with full stops and no following comma, as printed.
Why the others are wrong
- (a)6 and 30 — Neither number belongs to the sequence. 6 is not a Fibonacci number — the sequence runs 5, then 8, skipping it — and 30 is not one either, since the terms in that region are 21 and 34. The pair looks plausible only because both figures are of roughly the right size for a seventh and a tenth term, and roughly the right ratio apart. Any candidate who has actually written down the first ten terms can discard it in a second, which is exactly why writing them down is the first thing to do on a question of this kind rather than reasoning about positions in the abstract.
- (b)7 and 33 — The near-miss option, and the most dangerous of the three because both figures sit one below the correct ones. That is what makes it attractive to a candidate who has built the sequence correctly and then miscounted a position, or who has produced the right terms and mistrusted them. But 7 and 33 are not in the sequence: the terms around them are 5, 8, 21 and 34, and nothing between. Being one away from a correct value is not evidence of being nearly right when the answer must be a member of a specific set — it is evidence of an error somewhere in the construction.
- (d)10 and 39 — Again neither figure is a Fibonacci number, and this pair is the furthest above the answer. It would suit a candidate who has started counting from 1, 1 rather than from 0, 1, 1 and then estimated rather than computed — on that convention the seventh term is 13 and the tenth is 55, so even the alternative indexing does not produce 10 and 39. The option therefore corresponds to no consistent reading of the stem at all, and its function is to occupy the upper end of the numerical range so that the four options look like a graded scale.
Concept
The sequence in this question is generated by a RECURRENCE RELATION — a rule defining each term in terms of earlier ones, together with enough starting values to get it going. Here the rule is
a(n) = a(n-1) + a(n-2), with a(1) = 0 and a(2) = 1,
which is the Fibonacci sequence. Two starting values are needed because each term depends on two predecessors, and CHANGING THE STARTING VALUES CHANGES EVERY LATER TERM'S POSITION EVEN IF IT DOES NOT CHANGE THE SET OF VALUES. That is why a question of this type must state its indexing, and why the reader must find where it does so.
The Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 are worth memorising to about the twelfth term. Beyond appearing in puzzles, they carry two properties that questions use. The ratio of consecutive terms approaches the GOLDEN RATIO, roughly 1·618, so the terms grow geometrically rather than linearly — which is why a linear estimate of the tenth term from the seventh goes badly wrong. And the sequence appears in counting problems whose answer satisfies the same recurrence, such as the number of ways of climbing a staircase one or two steps at a time.
THE GENERAL METHOD for a sequence-construction item has three steps, and taking them in order is what makes such questions quick. ONE — extract the rule of formation and the starting values from the stem, in the stem's own terms and not from memory. TWO — write out the terms with their POSITIONS written above them. Positions in the head are where off-by-one errors live. THREE — read the answers off, and then check them against the option set. Where the answer must be a member of a set you have already generated, options containing non-members can be discarded outright.
That third step is a general one and applies well beyond sequences: whenever a question's answer must belong to a small computable set, generating the set first is usually faster than evaluating each option.
The quantitative strand of this paper includes several items about the structure of a sequence rather than about calculation, and this is the clearest of them. There is no arithmetic here worth the name — the largest sum required is 13 + 21 — and the entire content is whether the candidate reads the indexing instruction and applies it rather than the remembered convention.
That is why the stem is written the way it is. It could have said 'consider the Fibonacci sequence' in six words; instead it spends three sentences defining the rule from scratch and then adds a fourth sentence fixing which number is the first. The fourth sentence is the question. An examiner who defines a familiar object from scratch is usually signalling that the definition differs in some respect from the familiar one, and it is worth pausing wherever that happens.
The option set reinforces the point in an unusual way. Three of the four options contain no Fibonacci numbers at all, so the item is generous to a candidate who does any work on the page and unforgiving to one who tries to answer from memory of where the sequence reaches particular values. Generating the object and then testing the options against it is the reliable order of operations.
Two features of the printed page are worth naming: the ordinal endings of 7th and 10th are superscripts in the booklet, and the stem is set as an opening line, 'A certain sequence of integers is constructed as follows :', followed by an unindented paragraph.
Key facts
- With 0, 1 and 1 as the first, second and third terms, the sequence runs 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 — so the 7th term is 8 and the 10th is 34.
- The rule is the Fibonacci recurrence a(n) = a(n-1) + a(n-2), which needs two starting values because each term depends on two predecessors.
- On the more common convention beginning 1, 1, 2, 3, the 7th term is 13 and the 10th is 55; the stem fixes its own indexing precisely to exclude that reading.
- Fibonacci numbers to the twelfth term: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.
- None of 6, 7, 10, 30, 33 or 39 is a Fibonacci number, so three of the four options can be eliminated by generating the sequence alone.
- The ratio of consecutive Fibonacci numbers approaches the golden ratio, about 1·618, so the terms grow geometrically and cannot be estimated linearly.
- Changing the starting values of a recurrence shifts every later term's POSITION even when the set of values is unchanged.
Study next
Common traps
- Applying the remembered 1, 1, 2, 3 convention instead of the stem's 0, 1, 1. That gives 13 and 55, and neither pair is on the page.
- Counting positions mentally instead of writing them above the terms. Every error available in this question is an off-by-one error.
- Estimating the tenth term from the seventh. Fibonacci terms grow geometrically, so a linear estimate is badly wrong.
- Choosing a near-miss such as 7 and 33 on the strength of being close. The answer must be a member of the sequence, and neither figure is.
Sequence items on EPFO papers give the rule of formation in words and ask for a term at a stated position, or for a sum, or for the next term. They almost never require heavy arithmetic; the difficulty is placed in the reading — which number is the first, whether the count includes the starting values, whether a term is described by its position or by its value. Where the stem defines a familiar sequence from scratch instead of naming it, read the definition carefully, because the reason for defining it is usually that it differs from the standard form in some respect. Write the terms out with positions above them, and check the option set against the terms you have generated before deciding.
Related PYQs
EPFO_APFC_2016_Q37Numbers a1, a2, a3, a4, a5, ..., a24 are in arithmetic progression and a1 + a5 + a10 + a15 + a20 + a24 = 225. The value of a1 + a2 + a3 + a4 + a5 + ... a23 + a24 is
- (a) 525
- (b) 725
- (c) 850
- (d) 900
Answer(d) 900
The arithmetic progression item, where the rule of formation again supplies more than it appears to and the structure of the indices does the work.
EPFO_APFC_2016_Q97A palindrome is a number which reads the same from left as well as from right, for example, 23732. What is the number of palindromes between 10 and 1010 ?
- (a) 101
- (b) 100
- (c) 99
- (d) 90
Answer(b) 100
The palindrome item — another question whose answer must belong to a set the candidate can generate, so generating the set first beats testing the options one by one.
EPFO_APFC_2016_Q98In writing all the integers from 1 to 300, how many times is the digit 1 used ?
- (a) 160
- (b) 140
- (c) 120
- (d) 110
Answer(a) 160
The digit-counting item over the integers 1 to 300, which rewards the same discipline of enumerating systematically rather than estimating.
Practice
- practice — not a real PYQ
In the sequence 1, 1, 2, 3, 5, 8, ... where each term after the second is the sum of the two preceding terms, what is the ninth term ?
- (a)21
- (b)34
- (c)55
- (d)89
Answer(b) 34 — with 1 and 1 as the first two terms the sequence runs 1, 1, 2, 3, 5, 8, 13, 21, 34, so the ninth term is 34. Note how the indexing shifts the answer: on the convention that begins 0, 1, 1, the ninth term is 21 instead. The same values occupy different positions.
- practice — not a real PYQ
A staircase has 6 steps and can be climbed one or two steps at a time. In how many different ways can it be climbed ?
- (a)8
- (b)13
- (c)21
- (d)32
Answer(b) 13 — the number of ways of reaching step n is the number of ways of reaching step n-1 plus the number of ways of reaching step n-2, which is the Fibonacci recurrence. With 1 way for one step and 2 for two steps, the counts run 1, 2, 3, 5, 8, 13 for six steps.