How do we find the sum of n terms of the following?

Tr= 5^r/[(r+1)(r+2)]

I can see there's an arithmetic progression and a geometric progression?
How do we combine them both to find n terms.

It is neither arithmetic nor geometric. Take a look at the first few terms:

T1 = 5^1/(2*3) = 25/30
T2 = 5^2/(3*4) = 25/12
T3 = 5^3/(4*5) = 25/4
...

As wolframalpha shows, the general sum is not elementary:

http://www.wolframalpha.com/input/?i=sum+(r%3D1..n)+5%5Er%2F%5B(r%2B1)(r%2B2)%5D

I see! Thank you very much!

To find the sum of n terms of the given series, which consists of both an arithmetic progression and a geometric progression, we need to separate these two progressions and calculate their individual sums separately.

First, let's identify the arithmetic progression within the given series. In the given series, there is an arithmetic progression with the common difference of 1 starting from r = 0. We can rewrite the given series by separating this arithmetic progression as follows:

Tr = (5^r / [(r+1)(r+2)])
= {5^r / [r(r+1)]} - {5^r / [(r+2)(r+1)]}

The first term, {5^r / [r(r+1)]}, is the term from the geometric progression, and it follows the form of a standard geometric series. The second term, {5^r / [(r+2)(r+1)]}, is the term from the arithmetic progression.

Now, let's calculate the sum of the first term, which represents the geometric progression, for n terms. The sum of a geometric progression with the first term a, common ratio r, and n terms can be calculated using the formula:

S_geometric = a * (1 - r^n) / (1 - r)

Here, the first term of our geometric progression is a = 5^0 / (0)(0+1) = 1, and the common ratio is r = 1. Plugging these values into the formula, we get:

S_geometric = (1 * (1 - 1^n)) / (1 - 1)
= 1 * 0 / 0
= 0

Therefore, the sum of the terms from the geometric progression is zero.

Next, let's calculate the sum of the terms from the arithmetic progression. The arithmetic progression in this series starts from r = 0 and has the common difference of 1. The sum of an arithmetic progression with the first term a, common difference d, and n terms can be calculated using the formula:

S_arithmetic = (n / 2) * (2a + (n-1)d)

Here, the first term of our arithmetic progression is a = 5^0 / [(0+2)(0+1)] = 1/2, and the common difference is d = 1. Plugging these values into the formula, we get:

S_arithmetic = (n / 2) * (2*(1/2) + (n-1)*1)
= (n / 2) * (1 + n - 1)
= (n / 2) * n

So, the sum of the terms from the arithmetic progression is (n / 2) * n.

To find the sum of n terms of the given series, we add the sums of the geometric and arithmetic progressions:

Sum(n) = S_geometric + S_arithmetic
= 0 + (n / 2) * n
= (n^2 / 2)

Therefore, the sum of n terms of the given series Tr = 5^r / [(r+1)(r+2)] is (n^2 / 2).