n a certain arithmetic progression, the sum of the first and fifth terms is 18 and the fifth term is 6 more than thethird term. Find the sum of the first ten terms of the progression

To solve this problem, we need to use the formulas for the terms of an arithmetic progression (AP) and the sum of an AP.

Step 1: Understand the problem We know that in an AP, the difference between any two consecutive terms is constant. This difference is called the common difference (d). We also know that the sum of the first and fifth terms is 18 and the fifth term is 6 more than the third term.

Step 2: Formulate equations Let's denote the first term of the AP as a. Then the fifth term is a + 4d and the third term is a + 2d. From the problem, we have two equations:

a + a + 4d = 18 (since the sum of the first and fifth terms is 18) a + 4d = a + 2d + 6 (since the fifth term is 6 more than the third term)

Step 3: Solve the equations From the second equation, we can find the common difference d:

4d = 2d + 6 2d = 6 d = 3

Substitute d = 3 into the first equation:

2a + 4*3 = 18 2a + 12 = 18 2a = 6 a = 3

Step 4: Find the sum of the first ten terms The sum S of the first n terms of an AP is given by the formula S = n/2 * (2a + (n-1)d). Substituting a = 3, d = 3 and n = 10, we get:

S = 10/2 * (2*3 + (10-1)*3) = 5 * (6 + 27) = 5 * 33 = 165

So, the sum of the first ten terms of the progression is 165.