Find in terms of a & pi, the sum of the first 50 terms of the series: a+pi, a^2+2pi, a^3+3pi..where a is a fixed real number.
first 50 of geometric series with a = a and r = a
plus
first 50 of arithmetic series with a = pi and d = pi
https://www.purplemath.com/modules/series3.htm
To find the sum of the first 50 terms of the series: a+π, a²+2π, a³+3π, and so on, we can use the formula for the sum of a finite geometric series.
In this case, notice that each term can be written as a term of the form (a^k) + (k * π), where k represents the position of the term in the series.
Let's break down the given series and identify the terms:
Term 1: (a¹) + (1 * π)
Term 2: (a²) + (2 * π)
Term 3: (a³) + (3 * π)
...
We can observe that the general term of the series can be written as T(k) = (a^k) + (k * π), where k represents any positive integer.
Now, let's use the formula for the sum of a finite geometric series:
Sn = a * (1 - r^n) / (1 - r)
In our case, the first term (a+π) corresponds to a * (1 - r^1) / (1 - r), and the common ratio is r = a/π.
So, the general term of the series in terms of 'a' and 'k' can be written as T(k) = a * ((a/π)^k - 1) / ((a/π) - 1).
To find the sum of the first 50 terms, we substitute n = 50 into our formula for Sn:
S50 = a * ((a/π)^50 - 1) / ((a/π) - 1)
Therefore, the sum of the first 50 terms of the given series in terms of 'a' and 'π' is a * ((a/π)^50 - 1) / ((a/π) - 1).