Suav just started a running plan where he runs 20 miles the first week and then increases the number of miles he runs by \( 5 \% \) each week. If he keeps up this plan for 25 weeks, how many total miles would Suav have run, to the nearest whole number?

To find the total miles Suav runs over 25 weeks, we can represent the number of miles he runs each week as a geometric series. He starts by running 20 miles in the first week, and each subsequent week he increases his miles by 5%, which means he runs 105% of the previous week's mileage. The mileage for each week can be calculated as follows: - Week 1: \( 20 \) - Week 2: \( 20 \times (1 + 0.05) = 20 \times 1.05 = 21 \) - Week 3: \( 21 \times 1.05 = 21 \times 1.05 = 22.05 \) Continuing this, the number of miles run in week \( n \) can be expressed as: \[ \text{Miles in week } n = 20 \times (1.05)^{n-1} \] To find the total miles Suav runs over the 25 weeks, we calculate the sum of the geometric series: \[ S_n = a \frac{(1 - r^n)}{(1 - r)} \] Where: - \( a = 20 \) (the first term), - \( r = 1.05 \) (the common ratio), - \( n = 25 \). Calculating it: \[ S_{25} = 20 \frac{(1 - 1.05^{25})}{(1 - 1.05)} \] Now, let's calculate \( 1.05^{25} \): \[ 1.05^{25} \approx 3.386 \] Substituting this back in: \[ S_{25} = 20 \frac{(1 - 3.386)}{-0.05} = 20 \frac{-2.386}{-0.05} = 20 \times 47.72 = 954.4 \] Rounding to the nearest whole number, Suav would have run approximately **954 miles** over the 25 weeks.