Question

1. what is the sum of the first 10 powers of 2? assume the first term of the sequence is (a_1 = 2).

(a) (s_{20} = 2,100)
(b) (s_{10} = 1,550)
(c) (s_{10} = 2,046)
(d) (s_{20} = 1,000)

Explanation:

⚡ Using what you learned: Geometric Sequences

Step 1: Identify sequence parameters

The sequence consists of the first 10 powers of 2 starting with \( a_1 = 2 \):

$$ a_1 = 2^1 = 2 $$

$$ r = 2 $$

$$ n = 10 $$

Step 2: Apply the geometric series sum formula

$$ S_n = \frac{a_1(r^n - 1)}{r - 1} $$

$$ S_{10} = \frac{2(2^{10} - 1)}{2 - 1} $$

Step 3: Calculate the value

$$ S_{10} = 2(1024 - 1) $$

$$ S_{10} = 2(1023) = 2046 $$

Answer:

\( S_{10} = 2,046 \)