UPSC NDA Math Model Paper 6

Here, we have to find the sum of the series 5 + 55 + 555 + ..... upto n terms
i.e 5 + 55 + 555 + ...... to n terms
= 5 ⋅ [1 + 11 + 111 + ..... to n terms]
=

5

9

.[9+99+999+....... to n terms ]
=

5

9

.[(10−1)+(102−1)+(103−1) +......... to n terms ]
=

5

9

.[(10+102+103+........ to n terms )−n]
As we know that, Sn=a(

1−rn

1−r

) for |r|<1 and Sn=a(

rn−1

r−1

) for |r|>1
=

5

9

.[

10×(10n−1)

(10−1)

−n]
=

5

81

.(10n+1−9n−10)
Hence, the required sum is

5

81

.(10n+1−9n−10)