The system of equations reveals that the first term (a1) of the arithmetic progression is 7, and the number of terms (n) is 9, satisfying the given conditions.
We can use the formulas for the sum of an arithmetic progression (Sn), the nth term (an), and the general term (an = a_1 + (n - 1)d).
Given:
Sn = 245
d = 5
an = 47
First, let's use the formula for the sum of an arithmetic progression:
Sn = (n/2)[2a_1 + (n - 1)d]
Substitute the given values:
245 = (n/2)[2a_1 + (n-1) * 5]
Now, let's use the formula for the nth term:
an = a_1 + (n - 1)d
Substitute the given values:
47 = a_1 + (n - 1) * 5
Now, we have a system of two equations with two unknowns:
Equation 1: 245 = (n/2)[2a_1 + (n - 1) * 5]
Equation 2: 47 = a_1 + (n - 1) * 5
Let's solve this system of equations to find a1 and n. It might be easier to start with Equation 2:
47 = a_1 + 5n - 5
a_1 + 5n = 52
Now, substitute this into Equation 1:
245 = (n/2)[2(a_1 + 5n) + (n - 1) * 5]
245 = (n/2)[2a_1 + 15n + 5n - 5]
245 = (n/2)[2a_1 + 20n - 5]
245 = n[a_1 + 10n - 2.5]
Now, substitute the value of a1 + 5n:
245 = n * 52 - 2.5n^2
2.5n^2 - 52n + 245 = 0
Now, solve this quadratic equation for n. Once you find n, substitute it back into Equation 2 to find a1. To solve the quadratic equation 2.5n^2 - 52n + 245 = 0, you can use the quadratic formula:
n = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = 2.5, b = -52, and c = 245. Plug these values into the formula:
n = (52 ± √(2704 - 4 * 2.5 * 245)) / (2 * 2.5)
n = (52 ± √(2704 - 2450)) / 5
n = (52 ± √254) / 5
Now, calculate the two possible values for n:
n1 = (52 + √254) / 5 ≈ 9
n2 = (52 - √254) / 5 ≈ (52 - 16) / 5 ≈ 7.2
Since the number of terms (n) cannot be negative or in decimal, we discard n2 as a solution. Therefore, n = 9.
Now that we know n, we can substitute this back into Equation 2 to find a1:
a_1 + 5n = 52
a_1 + 5 * 9 = 52
a_1 + 45 = 52
a_1 = 7
So, the first term (a1) of the arithmetic progression is 7, and the number of terms (n) is 9.