Question

Treasure Hunt is an exciting and adventurous game where participants follow a series of clues/numbers/maps to discover hidden treasure. 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:

1. Which number is on first spot?     1
2. 
   
   1. Which spot is numbered as 112?     2
      OR
   2. What is the sum of all the numbers on the first 10 spots?    2
3. Which number is on the (n – 2)^th spot?     1

Answer 1

i. Put the value of n as 1

for n = 1

20 + 4n = 20 + 4(1) = 24

∴ The number on the first spot is 24.

ii. a. an = 112

20 + 4n = 112

4n = 92

n = 23

∴ The spot numbered as 112 is the 23rd spot.  

OR

b. S₁₀ = ${\displaystyle \frac{10}{2}[2\times 24+(10-1)\times 4]}$

= 5 [48 + 36]

= 5 × 84

S₁₀ = 420 

∴ The sum of all the numbers on the first 10 spots is 420.

iii. Put n = (n - 2)

a_n = 20 + 4n

= 20 + 4(n - 2)

= 20 + 4n - 8

= 12 + 4n

∴ The 12 + 4n number is on the (n - 2)^th spot.