(i) 5 + 55 + 555 + … to n terms.
Let us take 5 as a common term so we get,
5 [1 + 11 + 111 + … n terms]
Now multiply and divide by 9 we get,
5/9 [9 + 99 + 999 + … n terms]
5/9 [(10 – 1) + (102 – 1) + (103 – 1) + … n terms]
5/9 [(10 + 102 + 103 + … n terms) – n]
So the G.P is
5/9 [(10 + 102 + 103 + … n terms) – n]
By using the formula,
Sum of GP for n terms = a(rn – 1 )/(r – 1)
Where, a = 10, r = 102/10 = 10, n = n
a(rn – 1 )/(r – 1) =
(ii) 7 + 77 + 777 + … to n terms.
Let us take 7 as a common term so we get,
7 [1 + 11 + 111 + … to n terms]
Now multiply and divide by 9 we get,
7/9 [9 + 99 + 999 + … n terms]
7/9 [(10 – 1) + (102 – 1) + (103 – 1) + … + (10n – 1)]
7/9 [(10 + 102 + 103 + … +10n)] – 7/9 [(1 + 1 + 1 + … to n terms)]
So the terms are in G.P
Where, a = 10, r = 102/10 = 10, n = n
By using the formula,
Sum of GP for n terms = a(rn – 1 )/(r – 1)
7/9 [10 (10n – 1)/(10-1)] – n
7/9 [10/9 (10n – 1) – n]
7/81 [10 (10n – 1) – n]
7/81 (10n+1 – 9n – 10)
(iii) 9 + 99 + 999 + … to n terms.
The given terms can be written as
(10 – 1) + (100 – 1) + (1000 – 1) + … + n terms
(10 + 102 + 103 + … n terms) – n
By using the formula,
Sum of GP for n terms = a(rn – 1 )/(r – 1)
Where, a = 10, r = 10, n = n
a(rn – 1 )/(r – 1) = [10 (10n – 1)/(10-1)] – n
= 10/9 (10n – 1) – n
= 1/9 [10n+1 – 10 – 9n]
= 1/9 [10n+1 – 9n – 10]
(iv) 0.5 + 0.55 + 0.555 + …. to n terms
Let us take 5 as a common term so we get,
5(0.1 + 0.11 + 0.111 + …n terms)
Now multiply and divide by 9 we get,
5/9 [0.9 + 0.99 + 0.999 + …+ to n terms]
5/9 [9/10 + 9/100 + 9/1000 + … + n terms]
This can be written as
5/9 [(1 – 1/10) + (1 – 1/100) + (1 – 1/1000) + … + n terms]
5/9 [n – {1/10 + 1/102 + 1/103 + … + n terms}]
5/9 [n – 1/10 {1 - (1/10)n}/{1 – 1/10}]
5/9 [n – 1/9 (1 – 1/10n)]
(v) 0.6 + 0.66 + 0.666 + …. to n terms.
Let us take 6 as a common term so we get,
6(0.1 + 0.11 + 0.111 + …n terms)
Now multiply and divide by 9 we get,
6/9 [0.9 + 0.99 + 0.999 + …+ n terms]
6/9 [9/10 + 9/100 + 9/1000 + …+ n terms]
This can be written as
6/9 [(1 – 1/10) + (1 – 1/100) + (1 – 1/1000) + … + n terms]
6/9 [n – {1/10 + 1/102 + 1/103 + … + n terms}]
6/9 [n – 1/10 {1 - (1/10)n}/{1 – 1/10}]
6/9 [n – 1/9 (1 – 1/10n)]