First slide

Arithmetic progression

Question

If there are (2n + 1) terms in A.P., then the ratio of the sum of odd terms and the sum of even terms is

Moderate

A

$n : (n + 1)$
B

$(n + 1) : n$
C

$(n - 1) : n$
D

None of these

Solution

Let the A.P. containing (2n + 1) terms be
a, a + d, a + 2d, a + 3d, a + 4d, a + 5d, …, a + 2nd
The sum of odd terms of this A.P.
$\begin{aligned} = a + (a + 2 d) + (a + 4 d) + \dots to (n + 1) terms \\ = \frac{n + 1}{2} [2 a + (n + 1 - 1) \times 2 d] = (n + 1) (a + nd) \end{aligned}$
The sum of even terms of this A.P.
$\begin{array}{l} = (a + d) + (a + 3 d) + (a + 5 d) + \dots to n terms \\ = \frac{n}{2} [2 (a + d) + (n - 1) \times 2 d] = n (a + nd) \end{array}$
Hence, the required ratio = $\frac{n + 1}{n}$

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