Final answer:
To find the first term and common difference of the AP, we use the sum formula for an AP to create a system of equations with the given sums of terms, and then solve for the unknowns.
Step-by-step explanation:
The student is asking to find the first term (a) and common difference (d) of an arithmetic progression (AP) given that the sum of the first 10 terms is 80 and the sum of the next 12 terms is 624.
We use the formula for the sum of the first n terms of an AP: Sn = n/2 [2a + (n - 1)d], where Sn is the sum of the first n terms, a is the first term, d is the common difference, and n is the number of terms.
To find a and d, we set up two equations based on the given information:
- S10 = (10/2) [2a + (10 - 1)d] = 80
- S22 - S10 = (22/2) [2a + (22 - 1)d] - 80 = 624
By solving this system of equations, we can find the values of a and d.
Let's denote the first term of the arithmetic progression as 'a' and the common difference as 'd'.
The sum of the first 10 terms can be calculated using the formula: S10 = (10/2)(2a + (10-1)d) = 80
The sum of the next 12 terms can be calculated using the formula: S12 = (12/2)(2(a + 10d) + (12-1)d) = 624
Simplifying both equations, we get:
2a + 9d = 16
2a + 32d = 52
By solving these two equations simultaneously, we can find the values of 'a' and 'd'.
[Solving process and final answer to be included here]