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The sum \( S \) of an infinite geometric series is \( S = rac{a}{1-r} \).
Given \( S = 5 \) and \( a = 2 \), we have \( rac{2}{1-r} = 5 \).
Solving, \( 2 = 5(1-r) \), \( 2 = 5 - 5r \).
Rearranging, \( 5r = 3 \), so \( r = rac{3}{5} \).
#### \( rac{3}{5}\)1.
Calculate the sum of the first 15 positive even numbers.
The first 15 positive even numbers are: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, and 30.
Using the formula for the sum of the first \( n \) even numbers, \( S = n(n + 1) \), where \( n = 15 \):
\[ S = 15 imes (15 + 1) = 15 imes 16 = 240 \]
A rectangle has a length of \( 3x + 2 \) and a width of \( x - 1 \). If \( x = 5 \), what is the area of the rectangle?