The fifth term of an A.P is 23 and the twelfth is 37. Find the first term, the common difference and the sum of thefirst eleven terms

Sure, let's solve this step by step.

1. We know that the nth term of an arithmetic progression (A.P) is given by a + (n-1)d, where a is the first term and d is the common difference.

2. From the problem, we know that the 5th term is 23 and the 12th term is 37. So we can set up the following equations:

   a + 4d = 23 ...(equation 1)

   a + 11d = 37 ...(equation 2)

3. Subtract equation 1 from equation 2 to find the value of d:

   (a + 11d) - (a + 4d) = 37 - 23

   7d = 14

   d = 14/7 = 2

4. Substitute d = 2 into equation 1 to find the value of a:

   a + 4*2 = 23

   a = 23 - 8 = 15

5. So, the first term (a) is 15 and the common difference (d) is 2.

6. To find the sum of the first 11 terms (S11), we use the formula S_n = n/2 [2a + (n-1)d]:

   S11 = 11/2 [2*15 + (11-1)*2]

   S11 = 5.5 * [30 + 20] = 5.5 * 50 = 275

So, the first term is 15, the common difference is 2, and the sum of the first 11 terms is 275.