Question

The sum of first 8th term of an A.P is 100 and sum of first 19 terms is 551 find AP

Answer 1

Solution:-

Given

$\rm \to \: S_8 = 100$

$\rm \: \to \: S_{19} = 551$

Formula

$\boxed{ \red{ \rm \: S_n = \dfrac{n}{2} \{2a + (n - 1)d \}}}$

Now putting the value on formula

$\rm \: 100 = \dfrac{8}{2} \{2a + (8 - 1)d \}$

$\rm \: 100 = 4 \{2a + (7)d \}$

$\rm \: 25 = 2a + 7d \: \: \: \: \: \: \: ....(i)eq$

Now take

$\rm \: 551 = \dfrac{19}{2} \{2a + (19 - 1)d \}$

$\rm \: 1102 = 19 \{2a + 18d \}$

$\rm \: 58 = 2a + 18d \: \: \: \: \: \: \: \: ....(ii)eq$

Using elimination method

Subtract (i) from (ii) eq we get

$\rm \: 2a + 18d - (2a + 7d) = 58 - 25$

$\rm \: 2a + 18d - 2a - 7d = 33$

$\rm \: 11d = 33$

$\rm \: d \: = 3$

Now put the value of d on (i ) st equation

$\rm \: 25 = 2a + 7d \: \: \: \: \: \: \: ....(i)eq$

$\rm \: 25 = 2a + 7 \times 3$

$\rm \: 25 = 2a + 21$

$\rm \: 2a = 25 - 21$

$\rm \: \: 2a = 4$

$\rm \: a = 2$

So first term (a) = 2 and common difference (d) = 3

So series are

=> 2 , 5 , 8 , 11 , ............