0.2+0.22+0.022+ up to n terms is equal to

Step-by-step solution

The given series is S_n = 0.2 + 0.22 + 0.222 + up to n terms. We can write each term as T_k = 2 (0.1 + 0.01 + 0.001 + + 10^ -k ) = 2 _ i=1 ^ k 10^ -i . Alternatively, express the terms as T_k = 2 10 + 22 100 + 222 1000 + = _ k=1 ^ n 2 9 (1 - 10^ -k ) . Expanding the summation, we get S_n = 2 9 _ k=1 ^ n (1 - 10^ -k ) = 2 9 ( _ k=1 ^ n 1 - _ k=1 ^ n 10^ -k ) . The sum _ k=1 ^ n 1 = n . The sum _ k=1 ^ n 10^ -k is a geometric progression with first term a = 0.1 and common ratio r = 0.1 . Using the sum formula for a g