Find the sum of the series 2+11+101+1001+… to n terms.

The question seems to be incorrect it should be 11+101+1001+... up to n terms.

We can rewrite the given sesris as (10+1) + ( 100+1) + (1000+1)+... up to n terms.

(10+100+1000+... up to n terms) + ( 1+1+1+...+ up to n terms)

Now if we notice the series 10+100+1000+ ... up to n terms. is in G.P with a =10 and r = 10

hence S_n = a(rⁿ - 1)/r-1 

therefore Sn = 10 ( 10ⁿ -1)/(10-1) = 10 ( 10ⁿ -1)/9

hence  Sn = 10( 10ⁿ -1)/9

thus sum of the seseries 11+101+1001+... upto n terms = 10( 10ⁿ -1)/9 + n

Hope it helps you.