Q) 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 nth spot is 20 + 4n, then answer the following questions to help the players in spotting the clues:

(i) Which number is on first spot ?
(ii) Which spot is numbered as 112?
(iii) What is the sum of all the numbers on the first 10 spots?
(iv) Which number is on the (n – 2)th spot?

Ans: 

(i) Number is on first spot :

Given that number on n^th spot, T_n = 20 + 4n

∴ Number on first spot will be given by n = 1

By substituting n = 1 in the above expression, we get:

T₁ = 20 + 4 (1) = 24

Therefore, 24 is on first spot.

(ii) Spot numbered as 112:

Let’s consider n^th spot is numbered as 112

Therefore, 20 + 4n = 112

∴ 4n = 112 – 20 = 92

∴ n =

∴ n = 23

Therefore, 23rd spot is numbered as 112.

(iii) Sum of all the numbers on the first 10 spots:

To get the sum of first 10 spots, we need to get first term, a and common difference, d of the given AP.

We just calculated fist spot in part (i). ∴ a₁ = 24

Let’s calculate value of 2^nd spot, a₂ = 20 + 4 (2) = 28

and common difference, d = a₂  – a₁ = 28 – 24 = 4

Next, sum of n terms of an AP, S_n = [2 a + (n – 1) d]

We need to calculate sum of 10 terms, ∴ n = 10

∴ S₁₀ = [2 (24) + (10 – 1) (4)]

= 5 (48 + 36) = 5 x 84 = 420

Therefore, the sum of all the numbers on the first 10 spots is 420

(iv) Number on the (n – 2)^th spot:

It is given that the number on n^th spot, T₁ is given by 20 + 4 n

To get the value of the number on (n -2)^th spot, we substitute n = (n – 2) in the expression

∴ T_{n – 2} = 20 + 4 (n -2)

= 20 + 4 n – 8

= 12 + 4n
Therefore, number (12 + 4n) will be on (n-2)^th spot.

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