Problem: Given a geometric sequence with the first term î€ =0 and rî€ =0 and an arithmetic sequence with the first term =0. A third sequence 1,1,2,… is formed by adding corresponding terms of the two given sequences. The sum of the first ten terms of the third sequence is.
Answer Choices:
A. 978
B. 557
C. 467
D. 1068
E. not possible to determine from the information given.
Solution:
Let the two series be a,ar,ar2,⋯ and 0,d,2d,⋯,a+0=1;
∴a=1 and r+d=1,r2+2d=2.∴r=2,d=−1.
S1​=r−1a(rn−1)​=2−11(210−1)​=1023,
S2​=2n​[0+(n−1)d]=210​[0+9(−1)]=−45.
∴S=1023−45=978.