Question

Sum of the first 8termof an ap is 100 first 19tream is 551 find the sum of 25 terms




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Answer 1

Answer:

Sn = n/2 {2a + (n - 1) d}

S8 = 8/2 {2a + (8-1)d}

100 × 2 = 8{ 2a + 7d}

200÷ 8 = 2a + 7d

25 = 2a + 7d

2a + 7d = 25 ---> 1

S19 = 19/2 {2a + (19 - 1)d}

551 = 19/2 {2a + 18d }

551×2÷19 = 2a + 18d

2a + 18d = 58 ----> 2

solving 1 & 2

2a + 18d = 58

2a + 7d = 25

- - -

---------------

11d = 33

d = 33÷11

d = 3

substitute the value of d = 3 in equation 1

2a + 7d = 25

2a + 7(3) = 25

2a + 21 = 25

2a = 25 - 21

2a = 4

a = 4/2

a = 2

a25 = 25/2 {2(2) + (25 - 1)3}

a25 = 25/2 {4 + 24(3)}

a25 = 25/2 {76}

a25 = 25 × 38

a25 = 950

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Answer 2

GIVEN :–

• Sum of first 8 terms of an A.P. is 100.

• Sum of first 19 terms of an A.P. is 551.

TO FIND :–

• Sum of first 25 terms = ?

SOLUTION :–

• We know that sum of n terms is –

$\\ \: \: \bigstar \: \: \: \: \large { \green{ \boxed{ \bold{ S_{n} = \dfrac{n}{2} [ 2a + (n - 1)d ]}}}}$

• According to first condition –

$\\ \implies { \bold{ S_{8} = 100}}$

$\\ \implies { \bold{ \dfrac{8}{2}[2a + (8 - 1)d]= 100}}$

$\\ \implies { \bold{ 4(2a + 7d)= 100}}$

$\\ \implies { \bold{ 2a + 7d= 25 \: \: \: \: \: - - - - eq.(1)}}$

• According to second condition –

$\\ \implies { \bold{ S_{19} = 551}}$

$\\ \implies { \bold{ \dfrac{19}{2}[2a + (19 - 1)d]= 551}}$

$\\ \implies { \bold{ (2a + 18d)= 58}}$

$\\ \implies { \bold{ (25 - 7d)+ 18d= 58 \: \: \: \:[ \: using \: \: eq.(1)]}}$

$\\ \implies { \bold{ 25 + 11d= 58 }}$

$\\ \implies { \bold{ 11d= 58 - 25}}$

$\\ \implies { \bold{ 11d= 33 }}$

$\\ \implies \large{ \red{ \boxed{ \bold{ d= 3} }}}$

• Now put the value of 'd' in eq.(1) –

$\\ \implies { \bold{ 2a + 7(3)= 25 }}$

$\\ \implies { \bold{ 2a + 21= 25 }}$

$\\ \implies { \bold{ 2a = 25 - 21}}$

$\\ \implies { \bold{ a = \cancel \dfrac{4}{2} }}$

$\\ \implies \large{ \red{ \boxed{ \bold{ a= 2} }}}$

• So that , sum of first 25 terms is –

$\\ \implies{ \bold{ S_{25} = \dfrac{25}{2} [ 2(2) + (25 - 1)3 ]}}$

$\\ \implies{ \bold{ S_{25} = \dfrac{25}{2} [ 4 + 24 \times 3 ]}}$

$\\ \implies{ \bold{ S_{25} = \dfrac{25}{2} [ 4 +72]}}$

$\\ \implies{ \bold{ S_{25} = \dfrac{25}{2} (76)}}$

$\\ \implies{ \bold{ S_{25} = 25 \times 38}}$

$\\ \implies \large{ \red{ \boxed{ \bold{ S_{25} = 950}}}}$

Hence , Sum of first 25 terms = 950.