If an AP a - 1- Sn - S2n - Sn - constant- - n - N then the common difference d -

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Formula for Sum of n Terms of an AP

If an AP a = ...

Question

If an $A P a = 1, S n : (S_{2 n} - S_{n}) =$ constant, $\forall n \in N$ then the common difference $d = ?$

$4$

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$\frac{1}{2}$

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$2$

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$3$

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Solution

The correct option is C
$2$

Explanation for the correct option:
Step 1. Find the common difference $d$ :
Given, $a = 1$ and $S n : (S_{2 n} - S_{n}) =$ constant
$\frac{S_{1}}{(S_{2} - S_{1})} = \frac{S_{2}}{(S_{4} - S_{2})}$
$\Rightarrow$ $S_{1} (S_{4} - S_{2}) = S_{2} (S_{2} - S_{1})$
$\Rightarrow$ $S_{1} S_{4} - S_{1} S_{2} = {S_{2}}^{2} - S_{1} S_{2}$
$\Rightarrow$ ${S_{2}}^{2} = S_{1} S_{4}$
$\Rightarrow$ ${(a + a + d)}^{2} = a (\frac{4}{2}) (2 a + 3 d)$
$\Rightarrow$ ${(2 a + d)}^{2} = a (4 a + 6 d)$
Step 2. By Solving it, we get
$\begin{array}{rcl} ∴ d & = & 2 a \\ = & 2 (1) \\ = & 2 \end{array}$
Hence, Option ‘C’ is Correct.

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If an APa=1,Sn:(S2n-Sn)= constant, ∀n∈N then the common difference d=?

If an $A P a = 1, S n : (S_{2 n} - S_{n}) =$ constant, $\forall n \in N$ then the common difference $d = ?$