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NCERT Class XI Mathematics - Sequences and Series - Solutions

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Question : 19 of 106
Marks: +1, -0
In an A.P., if pth term is 1q\frac{1}{q} and qth term is 1p\frac{1}{p} , prove that the sum of first pq terms is 12\frac{1}{2} (pq + 1), where p ≠ q.
Solution:  
Let a be the first term & d be the common difference of the A.P., then
apa_p = a + (p - 1) d = 1q\frac{1}{q} ... (i)
aqa_q = a + (q - 1) d = 1p\frac{1}{p} ... (ii)
From (i) & (ii), we get
(p - 1) d - (q - 1) d = 1q−1p\frac{1}{q} - \frac{1}{p}
⇒ (p - 1 - q + 1) d = p−qpq\frac{p-q}{pq} ⇒ (p - q) d = p−qpq\frac{p-q}{pq} ⇒ d = 1pq\frac{1}{pq} ... (iii)
Then a + p−1pq\frac{p-1}{pq} = 1q\frac{1}{q} [From (i) & (iii)]
⇒ a = 1q−p−1pq\frac{1}{q} - \frac{p-1}{pq} = p−p+1pq\frac{p-p+1}{pq} = 1pq\frac{1}{pq}
Hence, the sum of first pq terms
SpqS_{pq} = pq2[2pq+pq−1pq]\frac{pq}{2}\left[\frac{2}{pq} + \frac{pq-1}{pq}\right] = pq2[2+pq−1pq]\frac{pq}{2}\left[\frac{2+pq-1}{pq}\right] = pq+12\frac{pq+1}{2} , where p ≠ q.
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