To find the sum of all integral powers of 2 between 1 and 100, we first need to identify the powers of 2 that lie in this range.
1. Start by listing the powers of 2:
- [tex]\( 2^0 = 1 \)[/tex]
- [tex]\( 2^1 = 2 \)[/tex]
- [tex]\( 2^2 = 4 \)[/tex]
- [tex]\( 2^3 = 8 \)[/tex]
- [tex]\( 2^4 = 16 \)[/tex]
- [tex]\( 2^5 = 32 \)[/tex]
- [tex]\( 2^6 = 64 \)[/tex]
- [tex]\( 2^7 = 128 \)[/tex] (This is greater than 100, so stop here)
2. Summing these powers of 2:
[tex]\[
1 + 2 + 4 + 8 + 16 + 32 + 64
\][/tex]
3. Calculate the sum step-by-step:
- [tex]\( 1 + 2 = 3 \)[/tex]
- [tex]\( 3 + 4 = 7 \)[/tex]
- [tex]\( 7 + 8 = 15 \)[/tex]
- [tex]\( 15 + 16 = 31 \)[/tex]
- [tex]\( 31 + 32 = 63 \)[/tex]
- [tex]\( 63 + 64 = 127 \)[/tex]
Therefore, the sum of all integral powers of 2 between 1 and 100 is [tex]\( 127 \)[/tex].
Based on the provided options, none of them match 127; however, the detailed step-by-step calculation confirms that the correct sum is indeed 127. The provided options might have an error or omission.
