Find the sum of the first 20 terms of an arithmetic progression, if its fifth term is 11 and its 16th term is 39.

Sure, let's solve this step by step.

Step 1: We know that the nth term of an arithmetic progression (AP) can be found using the formula: a + (n-1)d, where a is the first term, d is the common difference, and n is the term number.

Step 2: We can set up two equations using the information given in the problem. The 5th term is 11, so we have: a + 4d = 11. The 16th term is 39, so we have: a + 15d = 39.

Step 3: Subtract the first equation from the second to find the common difference: 11d = 28, so d = 28/11.

Step 4: Substitute d = 28/11 into the first equation to find the first term: a + 4*(28/11) = 11, so a = 11 - 4*(28/11) = 33/11.

Step 5: Now that we have the first term and the common difference, we can find the sum of the first 20 terms using the formula for the sum of an AP: S_n = n/2 * (2a + (n-1)d), where S_n is the sum of the first n terms.

Step 6: Substitute the values into the formula: S_20 = 20/2 * (2*(33/11) + (20-1)*(28/11)) = 10 * (6 + 532/11) = 10 * (66 + 532/11) = 5980/11 = 544.

So, the sum of the first 20 terms of the arithmetic progression is 544.