Question

The sum of the first five terms of an AP is 25 and the sum of its next five terms is -75 . Find the th 10 term of AP

Answer 1

Step-by-step explanation:

Answer is _40

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Answer 2

The value of 10th term of an AP is -23.

Step-by-step explanation:

We are given that the sum of first five terms of AP is 25 and sum of next five terms of AP is -75.

we have to find the 10th term of AP.

• Formula used,

$an=a+(n-1)*d$

'a' is the first term of AP, 'd' is the common difference and 'n' is the term number of AP.

For finding the 10th term of AP, first we have to find first term(a) and common difference(d).

• Calculation for 'a' and 'd'

We have the sum of first five terms of AP is 25 we can write it as,

$a1+a2+a3+a4+a5=25$          --(1)

and sum of next five terms of AP is -75, so

$a6+a7+a8+a9+a10=-75$        --(2)

By using formula

$an=a+(n-1)*d$

for example, $a1=a+(1-1)*d$

$a1=a$  because n is 1 here.

like this we can write equations (1) and (2) as,

$a+a+(2-1)*d+a+(3-1)*d+a+(4-1)*d+a+(5-1)*d=25$

$a+a+d+a+2d+a+3d+a+4d=25$

add similar terms,

$5a+10d=25$

and $a+2d=5$    ---(3)

from (2),

$a+(5-1)d+a+(6-1)d+a+(7-1)d+a+(8-1)d+a+(9-1)d+a+(10-1)d=-75$

$a+5d+a+6d+a+7d+a+8d+a+9d=-75$

$5a+35d=-75$

$a+7d=-15$      ---(4)

Now solving equations (3) and (4) by using Elimination Method.

extract a from equation (3),

$a=5-2d$

Substitute extracted 'a' in equation (4),

$5-2d+7d=-15$

$5d=-15-5$

$5d=-20$

$d=\frac{-20}{5}$

$d=-4$

substitute the value of d in extracted a,

$a=5-2d$

$a=5-2*(-4)$

$a=5+8$

$a=13$

Now we have the values of first term(a) and common difference(d) as 13 and -4 respectively.

• Calculation for 10th term of AP

by using formula we can write,

$a10=a+(10-1)*d$

       $=a+9d$

$d=-4$  and $a=13$

$a10=13+9*(-4)$

       $=13-36$

      $=-23$

The 10th term of AP is -23.