Let a be the first term and r be the common ratio of the G.P
According to the given condition,
a5 = a r 5–1 = a r4 = p … (1)
a8 = a r 8–1 = a r 7 = q … (2)
a11 = a r 11†“1 = a r 10 = s … (3)
Dividing equation (2) by (1), we obtain
\(\frac{ar7}{ar4}=\frac{q}{p}\)
r3 = \(\frac{q}{p}\) ... (4)
Dividing equation (3) by (2), we obtain
\(\frac{ar10}{ar7}=\frac{s}{q}\)
⇒ r3 =\(\frac{s}{q}\) ... (5)
Equating the values of r 3 obtained in (4) and (5), we obtain
\(\frac{q}{p}=\frac{s}{q}\)
⇒ q2 = ps
Thus, the given result is proved.
The sum $ 1 + \frac{1 + 3}{2!} + \frac{1 + 3 + 5}{3!} + \frac{1 + 3 + 5 + 7}{4!} + ... $ upto $ \infty $ terms, is equal to
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