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If the sum of the $3 r d$ and the $8 t h$ terms of an AP is 7 and the sum of the $7 t h$ and the $14 t h$ terms is –3, find the 10th term.

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-1

-2

-3

-4

answer is A.

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Detailed Solution

Given, the sum of the 3rd and 8th term of an AP is

a 3 + a 8 = 7

and the sum of 7th and 14th terms is

a 7 + a 14 = − 3

.
The nth terms of an A.P can be calculated by using the formula

a n = a + (n − 1) d

.
Let’s find the values of

a 3

and

a 8

.
Substitute n=3 in

a n = a + n − 1 d

a 3 = a + 3 − 1 d a 3 = a + 2 d

Similarly, when n = 8.

a 8 = a + 8 − 1 d a 8 = a + 7 d

.
Let’s substitute

a + 2 d

for

a 3

and

a + 7 d

for

a 8

a 3 + a 8 = 7

to obtain the simplified equation.

a 3 + a 8 = 7 a + 2 d + a + 7 d = 7 2 a + 9 d = 7 2 a = 7 − 9 d

.
Let’s obtain the equation for

a 7

, and

a 14

a 7 = a + 7 − 1 d a 7 = a + 6 d

a 14 = a + 14 − 1 d a 14 = a + 13 d

Substitute,

a + 6 d

for

a 7

and

a + 13 d

for

a n

a 7 + a n = − 3

to obtain the simplified equation.
So,

a 7 + a 14 = − 3 a + 6 d + a + 13 d = − 3 2 a + 19 d = − 3

Substitute,

7 − 9 d

for 2a in

2 a + 19 d = − 3

to obtain the value of d.
So,

2 a + 19 d = − 3 7 − 9 d + 19 d = − 3 10 d = − 3 − 7

Solve further,

10 d = − 10 d = − 1

Substitute, d=-1 in

2 a + 19 d = − 3

to obtain the value of

a

.
So,

2 a + 19 d = − 3 2 a + 19 − 1 = − 3 2 a − 19 = − 3 2 a = 19 − 3

Solve further,

2 a = 16 a = 162 a = 8

Substitute, -1 for d and 8 for a and 10 for n in

a n = a + n − 1 d

to obtain the value of

a 10

.
So,

a 10 = 8 + 10 − 1 − 1 = 8 + 9 − 1 = 8 − 9 = − 1

So, the tenth term is -1.
Therefore, the correct option is 1.

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