Question

5th term is 30 and 12th term is 65 find the 65th term

Answer 1

Answer :

$\large{\star\:\:\boxed{\bf{a_{65}\:=\:330}}\:\:\star}$

Explanation :

Given :–

• a₅ = 30
• a₁₂ = 65

To Find :–

• a₆₅ (65th Term of this A.P.)

Solution :–

We have ,

$\rightarrow\sf{a_5\:=\:30}$

$\rightarrow\sf{a_5\:=\:a\:+\:(5\:-\:1)d}$

$\rightarrow\sf{30\:=\:a\:+\:4d}\:\:-----\bf{(1)}$

We also have ,

$\rightarrow\sf{a_{12}\:=\:65}$

$\rightarrow\sf{a_{12}\:=\:a\:+\:(12\:-\:1)d}$

$\rightarrow\sf{65\:=\:a\:+\:11d}\:\:-----\bf{(1)}$

Subtracting Equation(2) from Equation(1) :-

$\rightarrow\sf{65\:-\:30\:=\:(a\:+\:11d)\:-\:(a\:+\:4d)}$

$\rightarrow\sf{35\:=\:a\:+\:11d\:-\:a\:-\:4d}$

$\rightarrow\sf{35\:=\:7d}$

$\rightarrow\sf{d\:=\:\dfrac{35}{7}}$

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

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

$\rightarrow\sf{30\:=\:a\:+\:4(5)}$

$\rightarrow\sf{30\:=\:a\:+\:20}$

$\rightarrow\sf{a\:=\:30\:-\:20}$

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

Now , we have

• a = 10
• d = 5
• n = 65

Putting these values in the Formula :

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

$\rightarrow\sf{a_{65}\:=\:10\:+\:(65\:-\:1)(5)}$

$\rightarrow\sf{a_{65}\:=\:10\:+\:(64\:\times\:5)}$

$\rightarrow\sf{a_{65}\:=\:10\:+\:320}$

$\rightarrow\boxed{\bf{a_{65}\:=\:330}}$

∴ The 65th Term of this A.P. is 330 .