Question

The sum of first 8 terms of an AP is 100 and sum of first 19 terms is 551 . find AP.​

Answer 1

hope it will help you

Answer 2

$\bold{For \: an \: AP}$

nth term = $\boxed{ \mathsf{a + (n-1)d}}$

Sum of nth terms = $\boxed{\mathsf{\dfrac{n}{2} \times (2a + (n-1)d)}}$

where,

First term = a

Common difference = d

$\large \bold{ACCORDING \: TO \: QUESTION }$

$\mathsf{t_8 = a + (8 -1)d}$

$\mathsf{100 = a + 7d}$----(1)

$\mathsf{t_19 = \dfrac{19}{2} \times (2a + (19)d)}$

$\mathsf{551= 9.5 (19 + 19d}$---(2)

$\mathsf{551 = 19a + 171d}$---(2)

Multiply equation (1) with 2

$\mathsf{100 = a + 7d}$ × 19

$\mathsf{1900 = 19a + 133d}$ -(1)

On subtracting (1) by (2)

$\mathsf{1900 = 19a + 133d}$

$\mathsf{551 = 19a + 171d}$

$\mathsf{1349 = 38d}$

$\mathsf{\dfrac{1349}{38} = d}$

$\huge \boxed {\mathsf{d = 35.5}}$

For a place value of d in any equation,

$\mathsf{100 = a + 7 \times 35.5}$

$\mathsf{100 - 248.5 = a }$

$\huge \boxed {\mathsf{a = - 184.5}}$

$\bold{For \: AP}$

$\mathsf{ a = -184.5}$

$\mathsf{ a + d = -184 + 35.5} = - 149$

$\mathsf{ a + 2d = -47 + 2 \times 35.5} =$

$\mathsf{ a + 3d = -184.5 + 3 \times 35.5 =78}$

$\mathsf{ a + 4d = -184.5 + 4 \times 35.5 =-42.5}$

$\bold{AP \: IS}$

$\huge \boxed {\mathsf{ - 184.5 , -149 ,-78 ,-42.5 ......}}$