Question

if the sum of the first five terms of an A.P. is 30 and the sum of first four term is 20 then the 50TH terms of the A.P. is​

Answer 1

Answer:

Step-by-step explanation:

Answer 2

Answer:

The value of 50th term is 100

Step-by-step explanation:

Given the first five terms of an A.P is 30

and the sum of first four terms is 20

we have to find the value of 50th term.

The sum of n terms of A.P then

$s_{n} =\frac{n}{2} (2a+(n-1)d)$

$a$ = first term

$n=$number of terms in the arithmetic progression

$d=$ common difference.

To find the $n^{th}$ term of an arithmetic progression

$a_{n} =a_{1} +(n-1)d$

$a_{1}$ is the first term

$d$ is the common difference

Given

$n=5\\s_{n} =30$

$30=\frac{5}{2} (2a+(5-1)d)\\30=\frac{5}{2} (2a+4d)$

Simplify the above equation

$30*2=5(2a+4d)\\60=10a+20d$

dividing both sides by 5

$12=2a+4d$    --------(1)

Given

$n=4\\s_{n}=20$

$20=\frac{4}{2} (2a+(4-1)d)\\20=2(2a+3d)$

Dividing both sides by 2

$10=2a+3d$      -------(2)

From equation(1) and equation (2)

$2a+4d=12\\2a+3d=10$

Subtracting equation(2) from equation(1)

$2a+4d-2a-3d=12-10\\d=2$

put this value in equation(2)

$2a+3(2)=10\\2a+6=10\\2a=10-6\\2a=4\\a=2$

Now we find the value of 50th term

that is $a_{50}$

$a_{50} =a+49d\\ =2+49(2)\\ =2+98\\ =100$