2024 CSAT — Q79

When a series isn’t an obvious arithmetic/geometric one, take differences — and use the known terms on both sides of the blank to lock the rule.

First differences: $14-3=11,\quad 39-14=25,\quad 84-39=45,\quad ?,\quad 258-?$.

Second differences of the known run: $25-11=14,\quad 45-25=20$ — increasing by $6$. Continue the second differences as $14, 20, 26, 32, \dots$:

• next first difference $= 45 + 26 = 71 \Rightarrow * = 84 + 71 = 155$,
• following first difference $= 71 + 32 = 103 \Rightarrow 155 + 103 = 258$ ✓ (matches the given last term).

The given final term $258$ confirms the pattern, so $* = 155$.

Cross-check (closed form). Each term is $n^3 + n(n+1)$: $n=1\!:1+2=3$, $n=2\!:8+6=14$, $n=3\!:27+12=39$, $n=4\!:64+20=84$, $n=5\!:125+30=155$, $n=6\!:216+42=258$. ✓

Answer: (b) $155$.

The series is cubic, so its second differences form an arithmetic progression ($14, 20, 26, 32$, step $6$) — that is the structural signature to recognise instead of hunting for a multiplier. The masterstroke is using the given $258$ as a check: a correct rule must regenerate it, which immediately rejects any candidate that only fits the left side. The closed form $n^3 + n(n+1)$ is the elegant confirmation, but in the exam the two-row difference table plus the $258$ checkpoint is the faster, safer route.