If a1, a2, ....., a50 are in G.P., then \(\frac{a_1-a_3+a_5-....+a_{4

1.	If a1, a2, ....., a50 are in G.P., then \(\frac{a_1-a_3+a_5-....+a_{49}}{a_2-a_4+a_5-....+a_{50}}\) is equal to(a) 0 (b) 1 (c) \(\frac{a_1}{a_2}\)(d) \(\frac{a_1}{a_{50}}\)
Answer» (c) \(\frac{a_1}{a_2}\) Let the first term of the G.P. be a and common ratio r. Then, \(\frac{a_1-a_3+a_5-....+a_{49}}{a_2-a_4+a_5-....+a_{50}}\) = \(\frac{a-ar^2+ar^4-....+ar^{48}}{ar-ar^3+ar^5-....+ar^{49}}\) = \(\frac{a(1-r^2+r^4-....+r^{48})}{ar(1-r^2+r^4-....+r^{48})}\) = \(\frac{a}{ar}\) = \(\frac{a_1}{a_2}\).

If a1, a2, ....., a50 are in G.P., then \(\frac{a_1-a_3+a_5-....+a_{49}}{a_2-a_4+a_5-....+a_{50}}\) is equal to(a) 0 (b) 1 (c) \(\frac{a_1}{a_2}\)(d) \(\frac{a_1}{a_{50}}\)

Answer»

(c) \(\frac{a_1}{a_2}\)

Let the first term of the G.P. be a and common ratio r. Then,

\(\frac{a_1-a_3+a_5-....+a_{49}}{a_2-a_4+a_5-....+a_{50}}\) = \(\frac{a-ar^2+ar^4-....+ar^{48}}{ar-ar^3+ar^5-....+ar^{49}}\)

= \(\frac{a(1-r^2+r^4-....+r^{48})}{ar(1-r^2+r^4-....+r^{48})}\) = \(\frac{a}{ar}\) = \(\frac{a_1}{a_2}\).