Question -
Answer -
(i) 1 – 1/3 + 1/32 –1/33 + 1/34 + … ∞
Given:
S∞ = 1– 1/3 + 1/32 – 1/33 + 1/34 + …∞
Where, a = 1, r = -1/3
By using the formula,
S∞ =a/(1 – r)
= 1 / (1 – (-1/3))
= 1/ (1 + 1/3)
= 1/ ((3+1)/3)
= 1/ (4/3)
= ¾
(ii) 8 + 4√2 + 4 + ….∞
Given:
S∞ = 8+ 4√2 + 4 + …. ∞
Where, a = 8, r =4/4√2 = 1/√2
By using the formula,
S∞ =a/(1 – r)
= 8 / (1 – (1/√2))
= 8 / ((√2 –1)/√2)
= 8√2 /(√2 –1)
Multiply and dividewith √2 + 1 we get,
= 8√2 /(√2 –1) × (√2 + 1)/( √2 + 1)
= 8 (2 + √2)/(2-1)
= 8 (2 + √2)
(iii) 2/5 + 3/52 +2/53 + 3/54 + …. ∞
The given terms can bewritten as,
(2/5 + 2/53 +…) + (3/52 + 3/54 + …)
(a = 2/5, r = 1/25)and (a = 3/25, r = 1/25)
By using the formula,
S∞ =a/(1 – r)
(iv) 10 – 9 + 8.1 – 7.29 +…. ∞
Given:
S∞ = 8+ 4√2 + 4 + …. ∞
Where, a = 10, r =-9/10
By using the formula,
S∞ =a/(1 – r)
= 10 / (1 – (-9/10))
= 10 / (1 + 9/10)
= 10 / ((10+9)/10)
= 10 / (19/10)
= 100/19
= 5.263