Question

In an AP, the sum of first 10 terms is -150 and the sum of next 10 terms is -550. Find the AP.

Answer 1

$Here \: is \: the \: answer \: of \: your \: question$

According to question we have given;

* Sum of first 10 terms is - 150

Means; S10 = - 150

Now,

Sn = n/2 [2a + (n-1)d]

S10 = 10/2 [2a + (10 - 1)d]

- 150 = 5 [2a + 9d]

- 30 = 2a + 9d ........(1)

Also, we have given that;

* Sum of next 10 terms is - 550

Means,

Sn = - 550

a = a + 10d

Sn = n/2 [2a + (n - 1)d]

- 550 = 10/2 [2 (a + 10d) + (10 - 1)d]

- 550 = 5 [2a + 20d + 9d]

- 550 = 5 [2a + 29d]

- 110 = 2a + 29d ........(2)

Now,

Solve (1) and (2) by elimination;

2a + 29d = -110
2a + 9d = - 30 (change it's sign)
_____________
0a + 20d = - 80
_____________

d = - 80/20

d = - 4

Put value of d in (1)

- 30 = 2a + 9(-4)

- 30 = 2a - 36

2a = 6

a = 6/2

a = 3

A.P. = a, a + d, a + 2d

= 3, 3 + (-4), 3 + 2(-4)

= 3, 3 - 4, 3 - 8

= 3, -1, -5......
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Answer 2

$\underline{\underline{\large{\mathfrak{Solution : }}}}$



$\textsf{Let,} \\ \\ \mathsf{\implies First \: term \: = \: a } \\ \\ \mathsf{\implies Common \: difference \: = \: d } \\ \\ \mathsf{\implies No. \: of \: terms \: = \: 10} \\ \\ \mathsf{\implies S_{10} \: = \: -150}$




$\underline{\textsf{According to question : }} \\ \\ \mathsf{\implies S_{10} \: = \: -150} \\ \\ \underline{\textsf{Using Formula : }} \\ \\ \boxed{\mathsf{\implies S_{n} \: = \: \dfrac{n}{2}\{2a \: + \: (n \: - \: 1)d \} }}$



$\mathsf{ \implies \dfrac{10}{2} \{2a \: + \: (10 \: - \: 1)d \} \: = \: - 150} \\ \\ \mathsf{ \implies5(2a \: + \: 9d) \: = \: - 150 } \\ \\ \mathsf{ \implies 2a \: + \: 9d \: = \: \dfrac{ - 150}{5} } \\ \\ \mathsf{ \implies 2a \: + \: 9d \: = \: - 30} \\ \\ \mathsf{ \therefore \quad 2a \: = \: - 30 \: - \: 9d \qquad...(1)}$



$\textsf{Now, sum of next 10 terms is -550.} \\ \\ \textsf{We can think of an A.P. whose first term} \\ \textsf{ is 11th term of previous A.P.}$



$\textsf{Now,} \\ \\ \mathsf{\implies First \: term \: = \: a \: + \: 10d } \\ \\ \mathsf{\implies Common \: difference \: = \: d } \\ \\ \mathsf{\implies No. \: of \: terms \: = \: 10} \\ \\ \mathsf{\implies S_{10} \: = \: -550}$



$\underline{\textsf{According to question : }} \\ \\ \mathsf{\implies S_{10} \: = \: -550} \\ \\ \underline{\textsf{Using Formula : }} \\ \\ \boxed{\mathsf{\implies S_{n} \: = \: \dfrac{n}{2}\{2a \: + \: (n \: - \: 1)d \} }}$



$\mathsf{ \implies \dfrac{10}{2} \{2(a \: + \: 10d) \: + \: (10 \: - \: 1)d \} \: = \: - 550} \\ \\ \mathsf{ \implies5(2a \: + \: 20d \: + \: 9d) \: = \: - 550 } \\ \\ \mathsf{ \implies5(2a \: + \: 29d) \: = \: - 550} \\ \\ \mathsf{ \implies 2a \: + \: 29d \: = \: \dfrac{ - 550}{5} } \\ \\ \mathsf{ \implies 2a \: + \: 29d \: = \: - 110 } \\ \\ \mathsf{ \therefore \quad 2a \: = \: - 110 \: - \: 29d \qquad...(2)}$




$\textsf{From (1) and (2) , we get : } \\ \\ \mathsf{\implies -30 \: - \: 9d \: = \: -110 \: - \: 29d} \\ \\ \mathsf{ \implies - 9d \: + \: 29d \: = \: - 110 \: + \: 30} \\ \\ \mathsf{ \implies20d \: = \: - 80 } \\ \\ \mathsf{ \implies d \: = \: \dfrac{ - 80}{20} } \\ \\ \mathsf{ \therefore \quad d \: = \: - 4}$



$\textsf{Plug the value of \textbf{d} in (1) : } \\ \\ \mathsf{\implies 2a \: = \: -30 \: - \: 9d } \\ \\ \mathsf{\implies 2a \: = \: -30 \: - \: 9(-4)} \\ \\ \mathsf{\implies 2a \: = \: -30 \: + \: 36} \\ \\ \mathsf{\implies 2a \: = \: 6 } \\ \\ \mathsf{\implies a = \: \dfrac{6}{2}} \\ \\ \mathsf{\therefore \quad a = \: 3 }$



$\textsf{The General Form of an A.P. is : } \\ \\ \textsf{a , ( a + d ) , ( a + 2d ) , ( a + 3d ) ,........} \\ \\ \mathsf{ 3 \: , \: ( 3 \: - \: 4 ) \: , \: ( 3 \: - \: 2 \: \times \: 4 ) \: , \: ( 3 \: - \: 3 \times \: 4 ) \: ,........} \\ \\ \mathsf{3 \: , \: -1 \: , \: ( 3 \: - \: 8 ) \: , \: ( 3 \: - \: 12 ) \: ,........} \\ \\ \mathsf{3 \: , \: -1 \: , \: -5 \: , \: -9 \: , .......}$




$\boxed{\textsf{The required A.P. is 3 , -1 , -5 , -9 , ......}}$