If the sixth term of an AP is 37.and the sum of the first six terms is 147 find the a)first term. b)sum of the first fifteen terms

a+5d = 37

6/2 (2a+5d) = 147
solve for a and d, and then get
S15 = 15/2 (2a+14d)

Did you know?

Did you know that in an arithmetic progression (AP), if you know the 6th term and the sum of the first six terms, you can find the first term and the sum of the first fifteen terms? In this case, the 6th term is given as 37 and the sum of the first six terms is 147. To find the first term, you can use the formula: First term (a) = 2 * 6th term - Sum of the first six terms = 2 * 37 - 147 = 74 - 147 = -73 Therefore, the first term is -73. To find the sum of the first fifteen terms, you can use the formula for the sum of an AP: Sum of n terms (S) = n/2 * (2a + (n - 1)d) In this case, n = 15 (as we are finding the sum of the first fifteen terms), a = -73 (the first term), and d is the common difference (which we do not have). However, since we are given the 6th term, we can find the common difference using the formula: 6th term = a + 5d Substituting the known values: 37 = -73 + 5d 5d = 37 + 73 5d = 110 d = 22 Now we can find the sum of the first fifteen terms: S = 15/2 * (2 * (-73) + (15 - 1) * 22) = 15/2 * (-146 + 14 * 22) = 15/2 * (-146 + 308) = 15/2 * 162 = 135 * 162 = 21,870 Therefore, the sum of the first fifteen terms is 21,870.