What should come in place of ‘?’ in the following series ? 117, 98, 80, 64, ?, 42
51
The missing term is 51 — option 3.
Step 1 — take the first differences.
| From | To | Difference |
|---|---|---|
| 117 | 98 | −19 |
| 98 | 80 | −18 |
| 80 | 64 | −16 |
These are not constant, so the series is not arithmetic.
Step 2 — take the differences of the differences.
\(-18-(-19)=+1,\qquad -16-(-18)=+2\)
The second differences are 1, 2 — so they should continue 3, 4. That gives the remaining first differences :
\(-16+3=-13,\qquad -13+4=-9\)
Step 3 — build the rest of the series.
| Term | 117 | 98 | 80 | 64 | ? | 42 |
|---|---|---|---|---|---|---|
| Difference | — | −19 | −18 | −16 | −13 | −9 |
\(64-13=51\qquad\text{and}\qquad51-9=42\)
The last step lands exactly on the 42 that the question already gives, which confirms the pattern.
The method to carry away. When first differences are irregular but shrinking or growing smoothly, take differences again. A series whose second differences form a simple run — here 1, 2, 3, 4 — is one of the commonest forms in these papers.
The value of the given last term. Because 42 is printed, the answer can be checked rather than merely guessed: any candidate answer x must satisfy both a plausible step from 64 and a plausible step to 42. Only 51 gives the smooth run −13, −9. Option 1, 48, would need steps of −16 and −6; option 2, 46, steps of −18 and −4; option 4, 55, steps of −9 and −13 — the right pair of numbers in the wrong order.
Hence, the answer is 51.
Similar Questions
- Find the next term: 7, 18, 41, 88, 183, ?
Important Questions from Mixed Series
Find the missing term in the following series:
2A11, 4D13, 12G17, 48J23, _______
A series is given, with one missing term. Choose the correct option from the given ones that will complete the sequence.
T9, V13, X17, Z21, B25, ?
Select the number from among the given options that can replace the question mark (?) in the following series.
6Q9, ____, 20Y23, 27C30
Find the next term.
D25G, H81I, L169P, P289S, ?
Find out the next term.
JL22, OQ32, TV42, ?
