Find the sum of the first fifty, n terms of the sequence (Xn) if Xn = 2n + 3.

Let us prove that this sequence is an arithmetic progression.

To do this, calculate the difference between the n + 1-th and n-th terms of this sequence:

хn + 1 – xn = 2 * (n + 1) + 3 – (2n + 3) = 2 * (n + 1) + 3 – 2n – 3 = 2n + 2 + 3 – 2n – 3 = 2.

Since the difference of n + 1-th and n-th members of this sequence is equal to the number 2 for any value of n, this sequence is an arithmetic progression with the difference d equal to 2.

We find the first term of this progression:

x1 = 2 * 1 + 3 = 5.

We calculate the sum of the first 50 terms of this sequence:

S50 = (2 * x1 + d * (50 – 1)) * 50/2 = (2 * x1 + d * 49) * 25 = (2 * 5 + 2 * 49) * 25 = (10 + 98) * 25 = 108 * 25 = 2700.

Answer: 2700.