Question

in an ap if Sn=210, Sn+1=253 find its common difference.

Answer 1

Answer:

I don't know exact answer but is it 43

Answer 2

The common difference between sum of n term and sum of (n+1) terms is 43  

Step-by-step explanation:

Given as :

For Arithmetic Progression

The sum of n term of an A.P = 210

The sum of n+1 term of an A.P = 253

For A.P , Sum of n terms = $S_n$

The sum of A.P for n terms

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

                                                                         d is common difference

i.e   210 = $\dfrac{n}{2}$ [ 2 a + ( n - 1 ) d ]  

or,  n  [ 2 a + ( n - 1 ) d ] = 210 × 2

Or,  n  [ 2 a + ( n - 1 ) d ] = 420                

Or,  2 a n + n²d - n d   = 420           ...........   1

Again

The sum of A.P for (n + 1) terms

$S_n_+_1$  = $\dfrac{n+1}{2}$  [ 2 a + ( n + 1 - 1 ) d ]  

Or, 253 = $\dfrac{n+1}{2}$  [ 2 a + n d ]

or,  ( n + 1 ) ( 2 a + n d ) = 253 × 2

i.e   ( n + 1 ) ( 2 a + n d ) = 506            

or, 2 a n + n²d + 2 a + n d = 506                           .............2

Solving eq 1and eq 2

(2 a n + n²d + 2 a + n d ) - ( 2 a n + n²d - n d) = 506 - 420

Or, (2 a n - 2 a n) + (  n²d  -  n²d  ) + ( 2 a + n d + n d ) = 86

or,   0 + 0 + (2 a + 2 n d ) = 86

Or,         a + n d = $\dfrac{86}{2}$

∴           a + n d = 43

So, The common difference between sum of n term and sum of (n+1) terms = a + n d = 43

Hence, The common difference between sum of n term and sum of (n+1) terms is 43  . Answer