Final answer:
To find the first term of the geometric sequence, we use the formula for the sum of the first n terms. For the monthly mortgage payment, we apply the annuity formula with inputs of principal, rate, and period. Lastly, the common difference in the arithmetic sequence is obtained by subtracting and then dividing by the position difference.
Step-by-step explanation:
Answer to Geometric Sequence Question
The sum of the first eight terms of a geometric sequence is given by the formula Sn = a(1 - rn) / (1 - r), where Sn is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms. Given that S8 = 1041.02, r = 0.46, and n = 8, we can solve for the first term a. Rearranging the formula to solve for a gives us a = Sn(1 - r) / (1 - rn). Substituting the known values and calculating gives us the value of the first term.
Answer to Mortgage Payment Calculation
The monthly mortgage payment can be calculated using the formula for an annuity: PMT = P [i / (1 - (1 + i)-n)], where PMT is the monthly payment, P is the principal amount, i is the monthly interest rate, and n is the total number of payments. Given a principal of $162,000, an annual interest rate of 3.875% compounded monthly (thus, i = 0.0325/12), and a 30-year loan period (which equals 360 months), we can compute the exact monthly mortgage payment.
Answer to Arithmetic Sequence Question
The common difference in an arithmetic sequence can be found by subtracting any two successive terms. In this case, if p35 = -9p + 4 and p28 = 5p - 17, first we find the difference between p35 and p28, and then divide by the difference in their positions (which is 7), revealing the value of the common difference d.