Question

Find the value of 3+33+333+.....…......n term

Answer 1

Sn = 3(1+11+111+..........n terms)

Multiply and divided by 9
= 3(1× 9/9 + 11×9/9 +111× 9/9+.......n terms)
= 3/9 (9+ 99 +999+........n terms)
= 3/9 [(10-1)+(100-1)+(1000-1).......n terms]
grouping add and subtract terms
= 3/9 [(10+100+1000) - (1+1+1+1)......n terms]
for the first set of term ,
a= 10, r= 10
for the second set of term,
a=1 ,r= 1
Sn = 3/9[a( r^n - 1)/r-1 - na]
= 3/9[10 (10^n - 1)/10-1 - n(1)]
= 3/9[10 (10^n -1)/9 -n]
= 3/9[10 (10^n-1) -9n/9]
=30 (10^n - 1) -9n/81