Q1. [4] § 5.4 Sum of First n Terms of an AP
Treasure Hunt is an exciting and adventurous game where participants follow a series of clues/numbers/maps to discover hidden treasures. Players engage in a thrilling quest, solving puzzles and riddles to unveil the location of the coveted prize.
While playing a treasure hunt game, some clues (numbers) are hidden in various spots collectively forming an A.P. If the number on the $n$th spot is $20 + 4n$, then answer the following questions to help the players in spotting the clues :
Based on the above, answer the following questions to help the players in spotting the clues :
- (i) Which number is on first spot ? [1]
- (ii) Which spot is numbered as 112 ? OR What is the sum of all the numbers on the first 10 spots ? [2]
- (iii) Which number is on the $(n - 2)$th spot ? [1]
Previously asked in CBSE board exam
2024 30/4/1 Q37
Generated by claude-sonnet-4-6 · 2026-06-15 10:31 · grounding stimulus
Model Answer
Given: $a_n = 20 + 4n$
(i) Number on the 1st spot:
$a_1 = 20 + 4(1) = 20 + 4 = \mathbf{24}$
(ii) OR Option 1: Which spot is numbered 112?
$20 + 4n = 112$
$4n = 92 \Rightarrow n = 23$
∴ 112 is on the 23rd spot.
OR Option 2: Sum of first 10 terms:
$a_1 = 24,\ a_{10} = 20 + 4(10) = 60$
$S_{10} = \dfrac{10}{2}(a_1 + a_{10}) = 5 \times (24 + 60) = 5 \times 84 = \mathbf{420}$
(iii) Number on the $(n-2)$th spot:
Replace $n$ with $(n-2)$:
$a_{n-2} = 20 + 4(n-2) = 20 + 4n - 8 = \mathbf{4n + 12}$
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Explanation
- Part (i): Simply substitute $n = 1$ in the given formula.
- Part (ii): Set $a_n = 112$ and solve for $n$ (Option 1), or use the sum formula $S_n = \frac{n}{2}(a_1 + a_n)$ (Option 2). Attempt only ONE option in the exam.
- Part (iii): Substitute $(n-2)$ in place of $n$ in the formula and simplify — a common 1-mark algebraic substitution step.
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