Question -
Answer -
Let us consider theLHS
(21/4 .41/8 .81/16. 161/32….∞)
This can be written as
21/4 .22/8 . 23/16 . 21/8 … ∞
Now,
21/4 + 2/8 + 3/16+ 1/8 + …∞
So let us consider 2x,where x = ¼ + 2/8 + 3/16 + 1/8 + … ∞ …. (1)
Multiply both sides ofthe equation with 1/2, we get
x/2 = ½ (¼ + 2/8 +3/16 + 1/8 + … ∞)
= 1/8 + 2/16 + 3/32 +… + ∞ …. (2)
Now, subtract (2) from(1) we get,
x – x/2 = (¼ + 2/8 +3/16 + 1/8 + … ∞) – (1/8 + 2/16 + 3/32 + … + ∞)
By grouping similarterms,
x/2 = ¼ + (2/8 – 1/8)+ (3/16 – 2/16) + … ∞
x/2 = ¼ + 1/8 + 1/16 +… ∞
x = ½ + ¼ + 1/8 + 1/16+ … ∞
Where, a = 1/2, r =(1/4) / (1/2) = 1/2
By using the formula,
S∞ =a/(1 – r)
= (1/2) / (1 – 1/2)
= (1/2) / ((2-1)/2)
= (1/2) / (1/2)
= 1
From equation (1), 2x =21 = 2 = RHS
Hence proved.