Question

If the sum of n terms of an A.P. is 3n² + 5n then which of its terms is 164?

Options

•  26th

•  27th

•  28th

• none of these.

Answer 1

Here, the sum of first n terms is given by the expression,

S_n = 3n² + 5n

We need to find which term of the A.P. is 164.

Let us take 164 as the n^th term

So we know that the n^thterm of an A.P. is given by,

a_n = S_n - S_n-1

So,

164 = S_n - S_n-1

164 = 3n² + 5n - [3(n-1)² +5(n-1) ] 

Using the property,

( a - b)² = a² + b² - 2ab

We get,

164 = 3n² + 5n - [3(n² + 1 - 2n) + 5 ( n-1)] 

164 = 3n² + 5n - [3n² + 3 - 6n + 5n - 5]

164 = 3n² + 5n -(3n² - n - 2)

164 = 3n² + 5n - 3n² + n + 2

164 = 6n + 2

Further solving for n, we get

6n = 164 - 2  

  `n = 162/6`

    n = 27

Therefore,  164 is the 27^th term of the given A.P.