**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:

T_{1} = 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). ∴ a_{1 } = 24

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

and common difference, d = a_{2 } – a_{1 } = 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

∴ S_{10} = [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, T_{1} 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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