c) \( 5 x ; 20 x ; 80 x ; \ldots \) is equal to \( 20480 x \) ?

To solve this, let's look at the pattern in the series: \( 5x, 20x, 80x, \ldots \). It appears that each term is being multiplied by 4 to get to the next term. Specifically, \( 5x \times 4 = 20x \) and \( 20x \times 4 = 80x \). Continuing this pattern, the next term would be \( 80x \times 4 = 320x \). If we keep multiplying by 4, we can keep generating terms. In terms of series, if you count how many times you multiply by 4, 20480x can indeed be reached, but we need to explicitly find out how many multiplications set us there. Calculating \( 5x \) multiplied by \( 4^n \) where \( n \) counts the number of times, we have: - For \( n=0 \): \( 5x \) - For \( n=1 \): \( 20x \) - For \( n=2 \): \( 80x \) - For \( n=3 \): \( 320x \) - For \( n=4 \): \( 1280x \) - For \( n=5 \): \( 5120x \) - For \( n=6 \): \( 20480x \) So, indeed \( 20480x \) is the seventh term of the series! Therefore, the complete series perfectly matches up to \( 20480x \).