find the sum of 1-1/2+1/4-1/8+...to n terms
Answers
Answer:
The terms of the series
1–1/2+1/4–1/8+……………
are in geometric progression as they decrease by a constant factor -1/2 that is readily obtained by dividing any term by that which immediately precedes it.
-1/2 ÷ 1 = -1/2
1/4 ÷ (-1/2) = -1/2
(-1/8) ÷ 1/4 =-1/2
and so on.
∴ the common ratio r = -1/2
The first term a = 1 . We have to find the sum of the first eight terms.
∴ n = 8 .
Now we use the formula for the sum of n number of terms in G. P.
S = a(1 - rⁿ)/(1 - r)
Substituting the values for a, r and n, the required sum to 8 terms is:
S₈ = 1 [1 - (-1/2)⁸]/[1 - (-1/2)] = [1 - (1/256)]/ (1 + 1/2)=(255/256) ÷ 3/2
= 255/256 x 2/3 = (3 x 5 x 17)/(128 x 3) = 85/128
Hence the required sum of the first eight terms of the given series is the proper fraction 85/128 (Answer) .
Verification of Answer:
S₈ =1/1 –1/2 +1/4 –1/8 +1/16 –1/32+ 1/64 –1/128
=(128–64+32–16+8–4+2–1)/128 = 85/128
Hence our answer is correct