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Q.
In the following two A.P. is how many terms are identical ? 2, 5, 8, 11.........to 60 terms 3, 5, 7, 9........terms
Sequences and Series
Solution:
$AP _{1}=2,5,8,11,14,17,21, \ldots . .60$ terms
$AP _{2}=3,5,7,9,11,13,15,17, \ldots .50$ terms
Since $AP _{2}$ contains less number of terms, the number of common terms is
decided by total number common of $AP _{2}$ with $AP _{1}$
From the APs we find that $2,5,8,11, . .$ terms of $AP _{2}$ are common with $AP _{1}$
$2,5,8,11,14, \ldots .50$
These are in $AP$ with first term 2 and last term 50 and common difference 3
Total number of terms $= n$
$n ^{\text {th }}$ term $a _{ n }= a _{1}+( n -1) d$
$50=2+( n -1) 3$
$48=( n -1) 3$
$16= n -1$
$n =17$
Hence, there are 17 terms common in both $APs$.