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 10th term of the AP.

Answer 1

Answer:

$a_{10}= -5.54$  

Step-by-step explanation:

Given that

$S_{first5} =25$

$S_{Next5} =-75$

As we know

$S_{n} =\frac{n}{2}(2a_{1}+(n-1)d)$

So

$S_{First5} =\frac{5}{2}(2a_{1}+(5-1)d)$

$25 =\frac{5}{2}(2a_{1}+4d)$

$50 =5(2a_{1}+4d)$

$10 =2a_{1}+4d$ .....(1)

Similarly

$S_{Nex5} =\frac{5}{2}(2a_{6}+(5-1)d)$

$-75 =\frac{5}{2}(2a_{6}+4d)$

$-150 =5(2(a_{1}+5d)+4d)$

$-30 =2a_{1}+10d+4d$  

$-30 =2a_{1}+14d$ .....(2)

Multiply equation (1) by -3 on both sides we get the following equation

$-30 =-6a_{1}-12d$ ....(3)

Comparing right sides of equation (2) and (3)

$2a_{1} +14d=-6a_{1}-12d$

Rearranging

$8a_{1}= -26d$

$a_{1}= - 3.25d$....(4)

Substituting value of  $a_{1}$ into equation (1)

$10 =2(-3.25d)+4d$

$10 =-6.5d+4d$

$10 =-10.5d$

$d=-.95$

Now, substitute the value of d into equation (4)

$a_{1}= - 3.25d$

$a_{1}= - 3.25(-0.95)$

$a_{1}= 3.1$

Now, lets find the 10th term

$a_{10}= a_{1} + 9d$

$a_{10}= }= 3.1+ 9(-0.952)$  

$a_{10}= -5.54$  

Answer 2

Answer:

-23

Step-by-step explanation:

Sum Of First 5 Terms=5/2(2a+4d)

Sum Of Next five Terms=S10-S5

-75=10/2(2a+9d)-25

-10=2a +9d

10=2a+4d

Therefore d= -4