If 3+5+7+⋯ upto n terms 5+8+11+⋯ upto 10 terms =7, then the value of n is

Given, 3+5+7+⋯+n terms 5+8+11+⋯+ to 10 terms =7⇒n2[3×2+(n−1)2]102[2×5+9×3]=7⇒n2[6+2n−2]5[10+27]=7⇒n2(2n+4)=7×5×37⇒n(n+2)=35×37⇒n(n+2)=35(35+2)∴n=35

If 3+5+7+⋯ upto n terms 5+8+11+⋯ upto 10 terms =7, then the value of n is

Given, 3+5+7+⋯+n terms 5+8+11+⋯+ to 10 terms =7⇒n2[3×2+(n−1)2]102[2×5+9×3]=7⇒n2[6+2n−2]5[10+27]=7⇒n2(2n+4)=7×5×37⇒n(n+2)=35×37⇒n(n+2)=35(35+2)∴n=35

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

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Detailed Solution

Given, $\frac{3 + 5 + 7 + \dots + n terms}{5 + 8 + 11 + \dots + to 10 terms} = 7$

$\begin{array}{lc} \Rightarrow & \frac{\frac{n}{2} [3 \times 2 + (n - 1) 2]}{\frac{10}{2} [2 \times 5 + 9 \times 3]} = 7 \\ \Rightarrow & \frac{\frac{n}{2} [6 + 2 n - 2]}{5 [10 + 27]} = 7 \\ \Rightarrow & \frac{n}{2} (2 n + 4) = 7 \times 5 \times 37 \\ \Rightarrow & n (n + 2) = 35 \times 37 \\ \Rightarrow & n (n + 2) = 35 (35 + 2) \\ ∴ & n = 35 \end{array}$