Question

the 4th term of an arithmetic progression is 3, and the 16th term is -45. find the 21st term​

Answer 1

AnsweR :

$\large{\dag\:\:\boxed{\star\:\:a_{21}\:=\:(-65)\:\:\star}\:\:\dag}$

ExplanatioN :

$\dagger$ GiveN :–

• a₄ = 3
• a₁₆ = (-45)

$\dagger$ To FinD :–

• a₂₁

$\dagger$ Formula ApplieD :–

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

$\dagger$ SolutioN :–

For finding a₂₁ we need to find a and d first.

We have ,

$\rightarrow\sf{a_4\:=\:3}$

$\rightarrow\sf{a_4\:=\:a\:+\:(4\:-\:1)d}$

$\rightarrow\sf{3\:=\:a\:+\:3d\:\:----(1)}$

and also ,

$\rightarrow\sf{a_{16}\:=\:(-45)}$

$\rightarrow\sf{a_{16}\:=\:a\:+\:(16\:-\:1)d}$

$\rightarrow\sf{(-45)\:=\:a\:+\:15d\:\:----(2)}$

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

$\rightarrow\sf{(-45)\:-\:(3)\:=\:a\:+\:15d\:-(a\:+\:3d)}$

$\rightarrow\sf{(-48)\:=\:a\:-\:a\:+15d\:-\:3d}$

$\rightarrow\sf{(-48)\:=\:12d}$

$\rightarrow\sf{d\:=\:(-\dfrac{48}{12} )}$

$\rightarrow\boxed{\sf{d\:=\:(-4 )}}$

Putting this value of 'd' in Equation(1) , we get :

$\rightarrow\sf{3\:=\:a\:+\:(3)(-4)}$

$\rightarrow\sf{3\:=\:a\:-\:12}$

$\rightarrow\sf{a\:=\:12\:+\:3}$

$\rightarrow\boxed{\sf{a\:=\:15}}$

Now , we have a = 15 , d = (-4) , n = 21 .

Putting these values in the formula :

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

$\rightarrow\sf{a_{21}\:=\:15\:+\:(21\:-\:1)(-4)}$

$\rightarrow\sf{a_{21}\:=\:15\:+\:[20\:\times\:(-4)]}$

$\rightarrow\sf{a_{21}\:=\:15\:+\:(-80)}$

$\rightarrow\boxed{\bf{a_{21}\:=\:(-65)}}$

∴ The 21st Term of this A.P. is (-65) .