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.

treat as two series , then take the sum of the two

geometric ... a, a^2, a^3, ... , a^49, a^50

arithmetic ... π, 2π, 3π, ... , 49π, 50π

To find the sum of the first 50 terms of the series, we need to find the expression for the general term first.

Let's observe the pattern:

First term: a + π
Second term: a^2 + 2π
Third term: a^3 + 3π
...
nth term: a^n + nπ

We can see that the nth term of the series is obtained by raising 'a' to the power of 'n' and then adding 'nπ'.

To find the sum of the first 50 terms, we can look for a formula that sums up all these terms.

The formula to find the sum of the first 'n' terms of a series is given by the arithmetic series formula:

S_n = (n/2)(a_1 + a_n)

where S_n is the sum of the first 'n' terms, 'a_1' is the first term, and 'a_n' is the nth term.

In this case, the first term is a + π, and the nth term is a^n + nπ.

Let's substitute these values into the formula:

S_50 = (50/2)((a + π) + (a^50 + 50π))

Simplifying further:

S_50 = 25(a + π) + (a^50 + 50π)

So, the sum of the first 50 terms in terms of 'a' and 'π' is 25(a + π) + (a^50 + 50π).

To find the sum of the first 50 terms of the series, we need to express each term in terms of 'a' and 'pi', and then add them up.

Let's observe the pattern in the terms:

First term: a + pi
Second term: a^2 + 2pi
Third term: a^3 + 3pi
...
nth term: a^n + npi

We can see that each term is obtained by adding the nth power of 'a' and multiplying it by the corresponding value of 'n' and then adding 'pi'.

Now, to find the sum of the first 50 terms, we need to add all the terms from the first to the 50th term. The nth term can be written as a^n + npi.

So, the sum of the first 50 terms can be written as:

(a^1 + 1pi) + (a^2 + 2pi) + (a^3 + 3pi) + ... + (a^50 + 50pi)

We can see that 'a' is a common factor in all the terms, so we can factor it out:

a (1 + a + a^2 + ... + a^49) + pi(1 + 2 + 3 + ... + 50)

Now, we need to find the sum of the first 50 natural numbers, which can be calculated using the formula for the sum of an arithmetic series: Sum = (n/2)(first term + last term), where 'n' is the number of terms.

Using this formula, the sum of the first 50 natural numbers is given by:

Sum = (50/2) * (1 + 50) = 25 * 51 = 1275

Now, we can substitute this value into our expression for the sum:

a (1 + a + a^2 + ... + a^49) + pi * 1275

Therefore, the sum of the first 50 terms of the given series, in terms of 'a' and 'pi', is:

(a^1 + a^2 + a^3 + ... + a^50) + 1275pi