Question

Find the $12$th term from the end of the following arithmetic progressions.
$3,5,7,9,.....201$

Answer 1

Given A.P

$3,5,7,9,...\dots ..,201$

first term of this A.P $a_1=3$

second term of this A.P $a_2=5$

common difference $d=a_2-a_1=5-3=2$

the nth term of an A.P is given by

$a_n=a_1+(n-1)d$

$a_n=3+(n-1)2$

$a_n=3+2n-2$

$a_n=1+2n\dots ..eq\left(1\right)$

first calculate number of terms(n) in this A.P

put $a_n=201$ the last term of A.P in above eq(1)

$201=1+2n$

$2n=201-1$

$n=\frac{200}{2}$

$⟹n=100$

hence this A.P contains $100$ terms

for calculating $12$th term from the end means $(100-11)$th term from starting of A.P

hence we have to calculate $89$th term of this A.P

so put $n=89$ in eq(1)

$a_{89}=1+2\times 89$

$a_{89}=1+178$

$a_{89}=179$

hence $12$th term of from the end of this A.P is $179$