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(A) The numerator is an arithmetic progression with first term $a_1 = 3$ and common difference $d_1 = 2$. The sum of $n$ terms is given by $S_n = \fra

Vedclass · Accepted Answer

$35$

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(A) The numerator is an arithmetic progression with first term $a_1 = 3$ and common difference $d_1 = 2$. The sum of $n$ terms is given by $S_n = \frac{n}{2}[2a_1 + (n-1)d_1] = \frac{n}{2}[6 + (n-1)2] = \frac{n}{2}[2n + 4] = n(n+2)$.The denominator is an arithmetic progression with first term $a_2 = 5$,common difference $d_2 = 3$,and $10$ terms. The sum is $S_{10} = \frac{10}{2}[2(5) + (10-1)3] = 5[10 + 27] = 5 	imes 37 = 185$.Given the ratio is $7$,we have $\frac{n(n+2)}{185} = 7$.$n^2 + 2n = 1295$.$n^2 + 2n - 1295 = 0$.Factoring the quadratic equation: $(n + 37)(n - 35) = 0$.Since $n$ must be positive,$n = 35$.

If $\frac{3 + 5 + 7 + \dots \text{ to } n \text{ terms}}{5 + 8 + 11 + \dots \text{ to } 10 \text{ terms}} = 7$,then the value of $n$ is

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