Question

If the sum of first 8 terms of an A.P is 88 and sum of first 13 terms is 273 then find the sum of first n' terms.

Answer 1

Answer:

The Sum of first n terms is 15 - 6 n  .

Step-by-step explanation:

Given as :

For A.P

The sum of first 8 terms = 88

∵  $S_n$ = $\dfrac{n}{2}$ [ 2 a + ( n - 1) d ]          where a = first term , d = common difference

So, $S_8$ =  $\dfrac{8}{2}$ × [ 2 a + ( 8 - 1) d ]  

or, 88 = 4 × ( 2 a + 7 d )  

Or, 88 = 8 a + 28 d

Or, 2 a + 7 d = 22                              ..........A

Again

The sum of first 13 terms = 273

∵ $S_n$ = $\dfrac{n}{2}$ [ 2 a + ( n - 1) d ]           where a = first term , d = common difference

So, 273 =  $\dfrac{13}{2}$ × [ 2 a + ( 13 - 1) d ]  

Or, 21 × 2 = 2 a + 12 d

or, 21 = a + 6 d

i.e  a + 6 d = 21                                ...........B

Solving eq A and eq B

( 2 a + 7 d ) - 2 × ( a + 6 d ) = 22 - 2 × 21

or, ( 2 a - 2 a ) + ( 7 d - 12 d ) = 22 - 42

Or, 0 - 5 d = - 20

∴        d = $\dfrac{20}{5}$

i.e d = 4

So, The common difference = d = 4

Put the value of d in eq B

a + 6 × 4 = 21

or, a + 24 = 21

Or, a = 21 - 24

∴   a = - 3

So, The first term = a = - 3

Again

Sum of first n terms =  $S_n$

i.e  $S_n$ = $\dfrac{-3}{2}$ [ 2 (-3) + ( n - 1) 4 ]          

Or, $S_n$  = $\dfrac{-3}{2}$ [ - 6 + 4 n - 4 ]

Or, $S_n$ = 9 - 6 n + 6

∴   $S_n$ = 15 - 6 n

So, The Sum of first n terms =  $S_n$ = 15 - 6 n

Hence, The Sum of first n terms is 15 - 6 n  . Answer

Answer 2

Step-by-step explanation:

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