Question

In an AP, if S_n = n(4n + 1), then find the AP.

Answer 1

S_n = n(4n + 1)

S_{n - 1} = (n - 1)[ 4(n - 1) - 1]

= (n - 1)[ 4n - 5]

S_n - S_{n - 1} = n(4n + 1) - (n - 1)(4n - 5)

(a₁ + a₂ + a₃ + - - - + a_{n - 1} + a_n) - (a₁ + a₂ + a₃ + - - - + a_{n - 1}) = 4n² + n - (4n² - 5n - 4n + 5)

a_n = 11n - 5

For a_1,

Put n = 1 so a₁ = 11(1) - 5 = 6

For a_2,

Put n = 2, so a₁ = 11(2) - 5 = 17

For a_1,

Put n = 3 so a₁ = 11(3) - 5 = 28

For a_1,

Put n = 4 so a₁ = 11(4) - 5 = 39

So AP is 6, 17, 28, 39, …