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Sum of a geometric progression with first term a and constant different r is given by:
sum_gp = a(1 - rn)/(1 - r)
Required sum is:
Sum =0.5 + 0.55 + 0.555 + ... [n terms]
= 5 (0.1 + 0.11 + 0.111 + ... [n terms])
= 5 (1/10 + 11/100 + 111/1000 + ... [n terms])
Multiply by 9/9 (= 1):
= 5/9 (9/10 + 99/100 + 999/1000 + ... [n terms])
= 5/9 [ (1 - 1/10) + (1 - 1/100) + (1 - 1/1000) + ... + (1 - 1/(10n) ]
= 5/9 [ (1 + 1 + 1 + ... [n terms]) - (1/10 + 1/100 + ... + 1/10n) ]
= 5/9 [ n - 1/10 (1 - (1/10)n) / (1 - 1/10) ]
= 5/9 [ n - 1/9 (1 - (1/10)n) ]
= 5/9 [ n - 1/9 (1 - 0.1n) ]
= 5/9 [ n - (10n - 1) / (9 x 10n) ]
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