## Geometric Series Worksheet Solution2

In the page geometric series worksheet solution2 you are going to see solution of each questions from the geometric series worksheet.

(4) Find the sum of the following finite series

(i) 1 + 0.1 + 0.01 + 0.001 + .......... + (0.1)⁹

Solution:

There are 10 terms in this series.

a = 1  r = 0.1/1       n = 10

= 0.1 < 1

sn = a(1- rn)⁄(1 - r)

S₁₀ = 1 [1- (0.1)^₁₀]/[1-0.1]

= [1- (0.1)^₁₀]/0.9

= [1- (0.1)^₁₀]/0.9

(ii) 1 + 11 + 111 + .............. to  20 terms

Solution:

There are 20 terms in this series. But this is not geometric series. To make it as geometric series we have to separate this into two series.For that first we have to multiply and divide the whole series by 9.

= 9/9[1 + 11 + 111 + .............. to  20 terms]

= 1/9[9 + 99 + 999 + .............. to  20 terms]

= 1/9[(10-1) + (100-1) + (1000-1) + .............. to  20 terms]

= 1/9[(10+100+1000+.............. to 20 terms)-(1+1+1+......to 20 terms)]

 a = 10     r = 100/10              r = 10 > 1 a = 1     r = 1/1            r = 1
sn = a(rn-1)⁄(r-1)
sn = n a

= 1/9 [10(10^₂₀ - 1)/(10-1) - 20 (1)]

= 1/9 [10(10^₂₀ - 1)/9 - 20]

= 10/81(10^₂₀ - 1) - 20/9

= 10/81(10^₂₀ - 1) - 20/9

(5) How many consecutive terms starting from the first term of the  series

(i) 3 + 9 + 27 + ........ would be 1092?

Solution:

1092 represents sum of n terms of the geometric series.From this we need to find the value of n.

Sn = 1092

a = 3    r = 9/3

r = 3 > 1

Sn = 3 [(3^n-1)]/(3-1)

1092 = (3/2) [(3^n-1)]

(1092 x 2)/3 = [(3^n-1)]

(1092 x 2)/3 = [(3^n-1)]

728 = 3^n-1

728 + 1 = 3^n

3^n = 729

3^n = 3⁶

n = 6

(ii) 2 + 6 + 18 + ........... would be 728?

Solution:

728 represents sum of n terms of the geometric series.From this we need to find the value of n.

Sn = 728

a = 2    r = 6/2

r = 3 > 1

Sn = 2 [(3^n-1)]/(3-1)

728 = 2 [(3^n-1)]/2

728 = (3^n-1)

728 + 1 = 3^n

3^n = 729

3^n = 3⁶

n = 6        geometric series worksheet solution2

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