Find the $ 20^{\text {th }} $ term of the AP whose $ 7^{\text {th }} $ term is 24 less than the $ 11^{\text {th }} $ term, first term being 12.
Given:
The \( 7^{\text {th }} \) term of an AP is 24 less than the \( 11^{\text {th }} \) term and the first term is 12.
To do:
We have to find the \( 20^{\text {th }} \) term.
Solution:
Let $a$ be the first term and $d$ be the common difference.
This implies,
$a_1=a=12$
$a_{7}=a+(7-1)d$
$=a+6d$........(i)
$a_{11}=a+(11-1)d$
$=a+10d$........(ii)
According to the question,
$a_7=a_{11}-24$
$a_{11}-a_7=24$
$a+10d-(a+6d)=24$ [From (i) and (ii)]
$10d-6d=24$
$4d=24$
$d=6$
Therefore,
$a_{20}=a+(20-1)d$
$=12+19(6)$
$=12+114$
$=126$
Hence, the \( 20^{\text {th }} \) term of the AP is 126.
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