onsider the infinite geometnc series: \( 45+40,5+36,45+\ldots \) Calculate the valuc of the TWELFTH term of the series (correct to TWO decimal places) Explain why this series converges. Calculate the sum to infinity of the series.

The series given can be rewritten as \( 45 + 40.5 + 36.45 + \ldots \). To find the twelfth term, we note that this is a geometric series where the first term \( a = 45 \) and the common ratio \( r = 0.9 \). The nth term of a geometric series can be calculated using the formula \( a_n = a \cdot r^{n-1} \). Thus, the twelfth term is \( a_{12} = 45 \cdot (0.9)^{11} \approx 45 \cdot 0.3138 \approx 14.16 \). This series converges because the absolute value of the common ratio \( |r| < 1 \). When \( r \) is less than one, the terms of the series get smaller and smaller, approaching zero, which ensures that the series will sum to a finite value rather than diverging to infinity. To calculate the sum to infinity of the series, we can use the formula for the sum of an infinite geometric series, \( S = \frac{a}{1 - r} \). Plugging in our values gives \( S = \frac{45}{1 - 0.9} = \frac{45}{0.1} = 450 \). So, the sum to infinity of the series is \( 450 \).