Question

fill in the blanks so that the resulting statement is true. the first four terms of \\(\sum\limits_{i=1}^{6} 2^i\\) are ____. the common ratio is ____. the first four terms of \\(\sum\limits_{i=1}^{6} 2^i\\) are \\(\boxed{}\\) the common ratio is \\(\boxed{}\\)

Explanation:

Step1: Find first term (i=1)

Substitute \( i = 1 \) into \( 2^i \): \( 2^1 = 2 \)

Step2: Find second term (i=2)

Substitute \( i = 2 \) into \( 2^i \): \( 2^2 = 4 \)

Step3: Find third term (i=3)

Substitute \( i = 3 \) into \( 2^i \): \( 2^3 = 8 \)

Step4: Find fourth term (i=4)

Substitute \( i = 4 \) into \( 2^i \): \( 2^4 = 16 \)

Step5: Calculate common ratio

Divide second term by first term: \( \frac{4}{2} = 2 \) (or \( \frac{8}{4} = 2 \), \( \frac{16}{8} = 2 \))

Answer:

The first four terms are \( 2, 4, 8, 16 \); the common ratio is \( 2 \)