In the series 1, 6, 15, 28, 45, ............ the next term will be :
2017
In the series 1, 6, 15, 28, 45, ............ the next term will be :
Answer: A. 66 — Concept: When the consecutive differences of a sequence are not constant but themselves change by a fixed amount, the sequence is quadratic in its position…
- A.
66
- B.
76
- C.
56
- D.
84
Attempted by 7 students.
Show answer & explanation
Correct answer: A
Concept: When the consecutive differences of a sequence are not constant but themselves change by a fixed amount, the sequence is quadratic in its position number. Any such sequence is continued in two moves: first extend the difference row by adding that fixed second difference to the last difference, then add the newly found difference to the last known term.
Application
Position | Term | Difference from previous term |
|---|---|---|
1 | 1 | — |
2 | 6 | 6 − 1 = 5 |
3 | 15 | 15 − 6 = 9 |
4 | 28 | 28 − 15 = 13 |
5 | 45 | 45 − 28 = 17 |
The first differences are 5, 9, 13 and 17, so the terms do not grow by a fixed amount and the sequence is not an arithmetic progression.
The second differences are 9 − 5 = 4, 13 − 9 = 4 and 17 − 13 = 4. This constant 4 is what makes the sequence quadratic, so the concept above applies.
Extend the difference row: the next difference is 17 + 4 = 21.
Add it to the last known term: 45 + 21 = 66. The next term is 66.
Cross-check
A constant second difference of 4 means the general term has the form Tn = 2n2 − n = n(2n − 1). Testing it: n = 1 gives 1, n = 2 gives 6, n = 3 gives 15, n = 4 gives 28 and n = 5 gives 45, which reproduces every listed term. For n = 6 it gives 6 × 11 = 66, matching the difference-row result, so the two independent routes agree.
These values 1, 6, 15, 28, 45, 66 are the hexagonal numbers; recognising the family lets you continue the sequence without recomputing the difference table each time.