given the geometric series: 15 + p + 5/3 + 5/9 + ...
2.2.1 determine the value of p.
2.2.1 calculate the sum of the 8 terms of the series.
2.2.3 explain why the series is convergent
2.2.1 To determine the value of p, we can see that the common ratio between consecutive terms is 1/p.
So, we can set up the following equation:
(1/p) = (5/3)/(p) = (5/9)/(5/3)
Solving this equation, we get p = 9/5.
2.2.2 To calculate the sum of the 8 terms of the series, we can use the formula for the sum of a geometric series:
S = a*(1 - r^n)/(1 - r)
where a = 15 (first term), r = 1/p = 1/(9/5) = 5/9 (common ratio), and n = 8 (number of terms).
Plugging in these values, we get:
S = 15*(1 - (5/9)^8)/(1 - 5/9)
S ≈ 15*(1 - 0.0016861)/(0.5555555)
S ≈ 15*0.9983139/0.5555555
S ≈ 27
Therefore, the sum of the 8 terms of the series is 27.
2.2.3 The series is convergent because the common ratio is less than 1 in absolute value (|5/9| < 1). When the common ratio is less than 1 in absolute value, the series converges to a specific value as the number of terms approaches infinity. In this case, the series will converge to a sum of 15/(1 - 5/9) = 27 as determined in the previous calculation.