Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. $\sum_{i = 1}^{10} 5 \cdot 2^{i}$

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Identify the terms of the geometric sequence. Here, the general term is given by $a_i = 5 \cdot 2^i$, where $i$ goes from 1 to 10.

Recognize that this is a geometric sequence with the first term $a_1 = 5 \cdot 2^1 = 10$ and common ratio $r = 2$ because each term is multiplied by 2 to get the next term.

Recall the formula for the sum of the first $n$ terms of a geometric sequence: $S_n = a_1 \cdot \frac{r^n - 1}{r - 1}$.

Substitute the known values into the formula: $n = 10$, $a_1 = 10$, and $r = 2$, so the sum is $S_{10} = 10 \cdot \frac{2^{10} - 1}{2 - 1}$.

Simplify the denominator and prepare to calculate the numerator $2^{10} - 1$ to find the sum $S_{10}$.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Geometric Sequence

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. For example, in the sequence 5, 10, 20, 40, the common ratio is 2. Understanding this helps identify the pattern in the given sum.

Sum of the First n Terms of a Geometric Sequence

The sum of the first n terms of a geometric sequence can be calculated using the formula S_n = a(1 - r^n) / (1 - r), where a is the first term and r is the common ratio. This formula simplifies adding many terms without computing each individually.

Index and Exponent in Summation Notation

Summation notation (Σ) represents the sum of terms indexed by i from a starting value to an ending value. In this problem, the exponent i in 2^i changes with each term, affecting the value of each term in the sum. Understanding how the index affects each term is crucial for applying the sum formula correctly.

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. ∑i=1105⋅2i\(\sum\)_{i=1}^{10} 5 \(\cdot\) 2^i