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8. Sequences and Series
easy
Consider an infinite $G.P. $ with first term a and common ratio $r$, its sum is $4$ and the second term is $3/4$, then
A
$a = \frac{7}{4},\,r = \frac{3}{7}$
B
$a = \frac{3}{2},\,r = \frac{1}{2}$
C
$a = 2,\,r = \frac{3}{8}$
D
$a = 3,\,r = \frac{1}{4}$
(IIT-2000)
Solution
(d) Here $\frac{a}{{1 – r}} = 4\,\,{\rm{and}}\,\,ar = \frac{3}{4}$.
Dividing these,
$r(1 – r) = \frac{3}{{16}}$or $16{r^2} – 16r + 3 = 0$
or $(4r – 3)(4r – 1) = 0$
$r = \frac{1}{4},\frac{3}{4}\,\,{\rm{and }}\ a = 3,\,1$
so $(a,r) = \left( {3,\frac{1}{4}} \right)\,,\,\left( {1,\frac{3}{4}} \right)$.
Standard 11
Mathematics
