The expression \(2^3 + 2^6 + 2^9 + 2^{12} + 2^{15}\) can be calculated using the formula for the sum of a geometric progression (GP). ### Steps: 1. **Identify the GP terms**: - The first term \(a = 2^3 = 8\). - The common ratio \(r = 2^3 = 8\) (since each term is \(2^3\) times the previous term). - There are 5 terms in the sequence. 2. **Use the Sum Formula for GP**: The sum \(S_n\) of the first \(n\) terms of a GP is: \[ S_n = a \cdot \frac{r^n - 1}{r - 1} \] where \(n = 5\). 3. **Substitute values**: \[ S_5 = 8 \cdot \frac{8^5 - 1}{8 - 1} \] 4. **Calculate**: - Calculate \(8^5\): \[ 8^5 = 32768 \] - Substitute \(8^5 = 32768\) into the formula: \[ S_5 = 8 \cdot \frac{32768 - 1}{7} = 8 \cdot \frac{32767}{7} = 8 \cdot 4681 = 37448 \] ### Answer: \[ 2^3 + 2^6 + 2^9 + 2^{12} + 2^{15} = 37448 \]
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The expression \(2^3 + 2^6 + 2^9 + 2^{12} + 2^{15}\) can be calculated using the formula for the sum of a geometric progression (GP).
### Steps:
1. **Identify the GP terms**:
- The first term \(a = 2^3 = 8\).
- The common ratio \(r = 2^3 = 8\) (since each term is \(2^3\) times the previous term).
- There are 5 terms in the sequence.
2. **Use the Sum Formula for GP**:
The sum \(S_n\) of the first \(n\) terms of a GP is:
\[
S_n = a \cdot \frac{r^n - 1}{r - 1}
\]
where \(n = 5\).
3. **Substitute values**:
\[
S_5 = 8 \cdot \frac{8^5 - 1}{8 - 1}
\]
4. **Calculate**:
- Calculate \(8^5\):
\[
8^5 = 32768
\]
- Substitute \(8^5 = 32768\) into the formula:
\[
S_5 = 8 \cdot \frac{32768 - 1}{7} = 8 \cdot \frac{32767}{7} = 8 \cdot 4681 = 37448
\]
### Answer:
\[
2^3 + 2^6 + 2^9 + 2^{12} + 2^{15} = 37448
\]