Question

if a1,a2,...,a50 are in GP dn what is (a1 - a3 + a5 - ... + a49 )/ (a2 - a4 + a6 - ... + a50)?

Answer 1

Answer:

$\bf{\frac{a_1-a_3+a_5 - ... +a_{49}}{a_2- a_4 + a_ - ... + a_{50}}=1/r}$

Step-by-step explanation:

$\text{Let}\:a_1=a\:\text{and r be the common ratio}$

$\text{Generl term}\:"a_n"=ar^{n-1}$

Now,

$\frac{a_1-a_3+a_5 - ... +a_{49}}{a_2- a_4 + a_ - ... + a_{50}}$

$=\frac{a-ar^2+ar^4 - ... +ar^{48}}{ar- ar^3 + ar^5- ... + ar^{49})}$

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

$=\frac{1}{r(1)}$

$=\frac{1}{r}$