Question

If 25th term of an AP is 245 and sum of first 10 term is 500 Find the value of its 100th term ​

Answer 1

Answer :

$\large{\boxed{\star\:\:a_{100}\:=\:995\:\:\star}}$

Explanation :

Given :–

• a₂₅ = 245
• S₁₀ = 500

To Find :–

• a₁₀₀ (100th term of this A.P.)

Formulae Applied :–

• $\boxed{\bf{\star\:\:a_n\:=\:a\:+\:(n\:-\:1)d\:\:\star}}$

• $\boxed{\bf{\star\:\:S_n\:=\:\dfrac{n}{2}\:[2a\:+\:(n\:-\:1)d] \:\:\star}}$

Solution :–

We have,

$\rightarrow\sf{a_{25}\:=\:245}$

$\rightarrow\sf{a_{25}\:=\:a\:+\:(25\:-\:1)d}$

$\rightarrow\sf{245\:=\:a\:+\:24d\:\:-----\bf{(1)}}$

Multiplying both sides by 2 :-

$\rightarrow\sf{2(245\:=\:a\:+\:24d)}$

$\rightarrow\sf{490\:=\:2a\:+\:48d\:\:-----\bf{(2)}}$

We also have ,

$\rightarrow\sf{S_{10}\:=\:500}$

$\rightarrow\sf{S_{10}\:=\:\dfrac{10}{2}\:[2a\:+\:(10\:-\:1)d] }$

$\rightarrow\sf{500\:=\:5\:\times\:(2a\:+\:9d) }$

$\rightarrow\sf{2a\:+\:9d\:=\:\dfrac{500}{5} }$

$\rightarrow\sf{2a\:+\:9d\:=\:100\:\:-----\bf{(3)} }$

Subtracting Equation(2) from Equation(3) :-

$\rightarrow\sf{2a+48d\:-\:(2a\:+\:9d)\:=\:490-\:100 }$

$\rightarrow\sf{2a+48d\:-\:2a\:-\:9d\:=\:390 }$

$\rightarrow\sf{39d\:=\:390 }$

$\rightarrow\sf{d\:=\:\dfrac{390}{39} }$

$\rightarrow\bf{d\:=\:10 }$

Putting this value of 'd' in Equation(1) :-

$\rightarrow\sf{245\:=\:a\:+\:24(10)}$

$\rightarrow\sf{245\:=\:a\:+\:240}$

$\rightarrow\sf{a\:=\:245\:-\:240}$

$\rightarrow\bf{a\:=\:5}$

Now , we have a = 5 , d = 10 and n = 100 .

Putting these values in the Formula :

$\rightarrow\sf{a_{n}\:=\:a\:+\:(n\:-\:1)d}$

$\rightarrow\sf{a_{100}\:=\:5\:+\:(100\:-\:1)(10)}$

$\rightarrow\sf{a_{100}\:=\:5\:+\:(99)(10)}$

$\rightarrow\sf{a_{100}\:=\:5\:+\:990}$

$\rightarrow\boxed{\bf{a_{100}\:=\:995}}$

∴ The 100th Term of this A.P. is 995 .