Problem: A line initially 1 inch long, grows according to the following law, where the first term is the initial length.
1+41​2​+41​+161​2​+161​+641​2​+641​+….
If the growth process continued forever, the limit of the length of the line is:
Answer Choices:
A. ∞
B. 34​
C. 38​
D. 31​(4+2​)
E. 32​(4+2​)
Solution:
Rearrange the terms:
​S1​=1+41​+161​+⋯=34​ and S2​=42​​+162​​+642​​+⋯=32​​∴S=S1​+S2​=31​(4+2​).​