Problem 1 :
If the first term of an infinite G.P. is 8 and its sum to infinity is 32/3 then find the common ratio.
Solution :
Since the given sequence is infinite series, the sum of the series = a/(1 - r)
a/(1-r) = 32/3
a = 8
8/(1-r) = 32/3
8(3) = 32(1-r)
24 / 32 = 1 - r
3/4 = 1 - r
r = 1 - (3/4) ==> 1/4
Problem 2 :
Find the sum to n terms of the series.
(i) 0.4 + 0.44 + 0.444 +........ to n terms
Solution :
0.4 + 0.44 + 0.444 +........ to n terms
= 4(0.1 + 0.11 + 0.111 + ...... to n terms)
= 4 ⋅ (9/9) (0.1 + 0.11 + 0.111 + ...... to n terms)
= (4/9)[0.9 + 0.99 + 0.999 +............n terms]
= (4/9)[(1 - 0.1) + (1 - 0.12) + (1 - 0.13) + .......n terms]
= (4/9)[(1+1+1........n terms) - [0.1+0.12+0.13 + .......n terms]
= (4/9)[n - 0.1 [1 - (0.1)n/(1-0.1)]
= (4/9)[n - 0.1 [1 - (0.1)n/0.9]
= (4/9)[n - [(1 - (0.1)n)/0.9]
(ii) 3 + 33 + 333 + ........... to n terms
Solution :
3 + 33 + 333 + ........... to n terms
= 3 [1 + 11 + 111 + ...........+ n terms]
= 3 ⋅ (9/9) (1 + 11 + 111 + ...... to n terms)
= (3/9)[9 + 99 + 999 +............n terms]
= (1/3)[(10-1) + (100 - 1) + (1000 - 1) +............n terms]
=(1/3)[(10+100+1000 + .......n terms)-(1+1+1+.........n terms)]
a = 10, r = 100/10 = 10, a = 1 and r = 1
= (1/3)[10(10n - 1)/(10-1))-n]
= (1/3)[(10/9)(10n - 1) - n]
= [(10/27)(10n - 1) - (n/3)]
Problem 3 :
Find the sum of the Geometric series :
3 + 6 + 12 +............+ 1536
Solution :
tn = 1536
a = 3, r = 6/3 = 2
arn-1 = 1536
3(2)n-1 = 1536
2n-1 = 1536/3
2n-1 = 512
2n-1 = 29
n - 1 = 9
n = 10
Now, we have to find the sum of 10 terms.
Sn = a(rn - 1)/(r - 1)
S10 = 3(210 - 1)/(2 - 1)
= 3(1024 - 1)
= 3(1023)
S10 = 3069
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