Consider the following statements I. If an denotes the nth term of an AP, then an=an+k+an−k2 II. In an AP, if the Sum of m terms is equal to the sum of n terms, then the sum of(m + n) terms is always zero. Which of the statement is given above is/are correct?

A
Only I
B
only II
C
Both I and II
D
None of these

Solution:

Statement I Since, nth term of an AP is $α_{n}$ . $∴ a_{n} = \frac{a_{n + k} + a_{n - k}}{2} [∵ AM = \frac{a + b}{2}]$ Statement II Let $a$ be the first term and d be the common difference of an AP.

According to the given condition,

$\begin{array}{l} S_{m} = S_{n} \\ ∴ \frac{m}{2} [2 a + (m - 1) d] = \frac{n}{2} [2 a + (n - 1) d] \\ \Rightarrow (m - n) 2 a + (m^{2} - n^{2} - m + n) d = 0 \\ \Rightarrow (m - n) [2 a + (m + n - 1) d] = 0 . . . (i) \\ ∴ S_{(m + n)} = \frac{m + n}{2} [2 a + (m + n - 1) d] = 0 \\ = \frac{m + n}{2} (0) [∵ from eq . (i)] \\ = 0 \end{array}$

Consider the following statements $I. If a_{n}$ denotes the nth term of an AP, then $a_{n} = \frac{a_{n + k} + a_{n - k}}{2}$ II. In an AP, if the Sum of m terms is equal to the sum of n terms, then the sum of(m + n) terms is always zero. Which of the statement is given above is/are correct?

Solution:

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