Question

In an A.P., the sum of first n terms is (3n²/2)+(13n/2). Find its 25th term.

Answer 1

Answer:

The 25th terms of an Arithmetic progression is 85

Step-by-step explanation:

Given as :

The sum of first n terms of an AP is $\dfrac{3n^{2} }{2}$  +  $\dfrac{13n}{2}$

For Arithmetic progression

Sum of n term = $S_n$ = $\dfrac{n}{2}$ [ 2 a + ( n - 1 ) d ]

∵  $S_n$  =  $\dfrac{3n^{2} }{2}$  +  $\dfrac{13n}{2}$

i.e $S_n$ = $\dfrac{3n^{2}+13n }{2}$

Or, $S_n$ = $\dfrac{n}{2}$ (3 n + 13)

And

$S_n_-_1$  =  $\dfrac{3(n-1)^{2} }{2}$  +  $\dfrac{13(n-1)}{2}$

Or,  $S_n_-_1$ = $\dfrac{3(n^{2}+1-2n)+13(n-1) }{2}$

Or, $S_n_-_1$ = $\dfrac{3n^{2}+3-6n+13n-13 }{2}$

Or, $S_n_-_1$ = $\dfrac{3n^{2}+7n-10 }{2}$

Again

∵  $a_n$  = $S_n$  -  $S_n_-_1$

i.e  $a_n$ = ( $\dfrac{3n^{2}+13n }{2}$  ) -  ( $\dfrac{3n^{2}+7n-10 }{2}$  )

Or, $a_n$ =  $\dfrac{3n^{2}-3n^{2} +13n - 7n+10 }{2}$

∴  $a_n$ = $\dfrac{6n+10}{2}$

i.e $a_n$ = 3 n + 5

So, nth terms of an A.P = $a_n$ = 3 n + 5

Now, for n = 25

$a_2_5$  = 3 × 25 + 10

$a_2_5$  = 75 + 10

∴ $a_2_5$ = 85

25th terms of A.P = 85

So, The 25th terms of an Arithmetic progression =  $a_2_5$ = 85

Hence, The 25th terms of an Arithmetic progression is 85 . Answer