Final answer:
In an arithmetic progression (A.P.), the sum of the first and fifth terms is given as 5, and the sum of the twelfth and sixteenth terms is given as 15. Using the formulas for the sum of an arithmetic progression and solving the resulting equations, the sum of the first sixteen terms is 240.
Step-by-step explanation:
In an arithmetic progression (A.P.), each term is obtained by adding a constant difference to the previous term. Let the first term be a and the common difference be d. The sum of the first and fifth terms is given as 5, which can be expressed as a + 4d = 5. Similarly, the sum of the twelfth and sixteenth terms is 15, which can be expressed as a + 11d + a + 15d = 15.
Simplifying the equations, we get 2a + 19d = 15 and a + 4d = 5. Solving these equations simultaneously, we find a = -2 and d = 1. The formula for the sum of an arithmetic progression is S = (n/2)(2a + (n-1)d), where S is the sum, n is the number of terms, a is the first term, and d is the common difference.
Substituting the values, we get S = (16/2)(2(-2) + (16-1)1) = 8(30) = 240.