Question
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 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
Answers
Answered by
GPT 3.5
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.
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.