Find the 25^{th} common term of the following | vedclass

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(A) For $S_1$,the first term $a_1 = 1$ and common difference $d_1 = 6 - 1 = 5$. The general term is $T_n = 1 + (n-1)5 = 5n - 4$.For $S_2$,the first te

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$491$

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(A) For $S_1$,the first term $a_1 = 1$ and common difference $d_1 = 6 - 1 = 5$. The general term is $T_n = 1 + (n-1)5 = 5n - 4$.For $S_2$,the first term $a_2 = 3$ and common difference $d_2 = 7 - 3 = 4$. The general term is $T_m = 3 + (m-1)4 = 4m - 1$.For common terms,$5n - 4 = 4m - 1$,which implies $5n = 4m + 3$.The first common term is $11$ (where $n=3, m=3$).The common difference of the new sequence formed by common terms is $LCM(d_1, d_2) = LCM(5, 4) = 20$.The sequence of common terms is $11, 31, 51, .....$ which is an $A.P.$ with first term $A = 11$ and common difference $D = 20$.The $25^{th}$ common term is $A_{25} = A + (25-1)D = 11 + 24 	imes 20 = 11 + 480 = 491$.

Find the $25^{th}$ common term of the following $A.P.'s$:
$S_1 = 1, 6, 11, .....$
$S_2 = 3, 7, 11, .....$

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Find the $25^{th}$ common term of the following $A.P.'s$:$S_1 = 1, 6, 11, .....$$S_2 = 3, 7, 11, .....$