The average of seven two-digit numbers which are in arithmetic progression is 37. If the greatest of them is left out, the average reduces by 1. If the smallest of them is left out, the average increases by 1. Find the common difference of the arithmetic progression.1234

Sure, let's solve this step by step.

Step 1: Understand the problem We have seven two-digit numbers in an arithmetic progression (AP). The average of these numbers is 37. If we remove the greatest number, the average reduces by 1 (becomes 36). If we remove the smallest number, the average increases by 1 (becomes 38).

Step 2: Formulate the equations Let's denote the first term of the AP as 'a' and the common difference as 'd'. Since there are seven terms, the fourth term will be the average (a + 3d = 37).

When the greatest term is removed, the average of the remaining six numbers is 36. So, the sum of these six numbers is 636 = 216. The sum of all seven numbers is 737 = 259. Therefore, the greatest term (seventh term) is 259 - 216 = 43.

When the smallest term is removed, the average of the remaining six numbers is 38. So, the sum of these six numbers is 6*38 = 228. Therefore, the smallest term (first term) is 259 - 228 = 31.

Step 3: Solve the equations We now have two equations: a + 3d = 37 (from the average of all seven numbers) a = 31 (from the smallest term)

Substituting a = 31 into the first equation gives us 31 + 3d = 37, which simplifies to 3d = 6, so d = 2.

Therefore, the common difference of the arithmetic progression is 2.