NCERT Class XI Mathematics - Sequences and Series - Solutions

Question : 79

Total: 106

Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.

Solution:

Let the first term be a and the common difference be d of the A.P., then
am+n = a + [(m + n) – 1] d .....(i), am – n = a + [(m – n) – 1]d .....(ii)
Adding (i) & (ii), we get
am+n+am–n = a + [(m + n) – 1] d + a + [(m – n) – 1] d = 2a + [m + n – 1 + m – n – 1]d
= 2a + [2m – 2]d = 2a + 2 (m – 1)d = 2[a + (m – 1) d] = 2 am.
Hence, am+n+am–n = 2am.

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