Given an arithmetic sequence with a7 19 and a12 54 Find a1

Given an arithmetic sequence with a_7 = 19 and a_12 = 54. Find a_15. 61 68 75 82

Solution

a₇ = 19 and a₁₂ = 54. Find a_15.

In arithmetic sequence, we know the formula:

a_n = a + (n-1)d

a₇ = a + (7-1)d   

19 = a + 6d ----------------(1)

Now put a₁₂:

a₁₂ = a +(12-1)d

54 = a + 11d -------------- (2)

Solving equation (1) and (2) for a and d, we get:

a = - 23 and d = 7

Therefore, a₁₅ = - 23 + (15-1)*7

a₁₅ = - 23 + 14*7

a₁₅ = -23 + 98

a₁₅ = 75