Seatwork: Solve the following: 1. What is the 18th term of the arithmetic sequence 5, 8, 11, 14, ... 2.Which term of the arithmetic sequence is 30 given that a1 = 4 and d = 2? 3. How many terms does the arithmetic sequence have with d = - 26 and -19 as first and last term respectively?
Answer (1)
Answer: Problem 1: Find the 18th term of the arithmetic sequenceGiven:- First term (a₁) = 5- Common difference (d) = 8 - 5 = 3Solution:1. Use the formula for the nth term of an arithmetic sequence:aₙ = a₁ + (n - 1)d2. Plug in the values:a₁₈ = 5 + (18 - 1)3a₁₈ = 5 + 17(3)a₁₈ = 5 + 51a₁₈ = 56Answer: The 18th term of the arithmetic sequence is 56.Problem 2: Find the term number of the arithmetic sequenceGiven:- First term (a₁) = 4- Common difference (d) = 2- nth term (aₙ) = 30 Solution:1. Use the formula for the nth term of an arithmetic sequence:aₙ = a₁ + (n - 1)d2. Plug in the values:30 = 4 + (n - 1)230 - 4 = (n - 1)226 = (n - 1)2(n - 1) = 26/2(n - 1) = 13n = 14Answer: The term number of the arithmetic sequence is 14.Problem 3: Find the number of terms in the arithmetic sequenceGiven:- First term (a₁) = -19- Last term (aₙ) = -26- Common difference (d) = unknown, but we can find n using the formulaSolution1. Use the formula for the nth term of an arithmetic sequence:aₙ = a₁ + (n - 1)d2. We don't know d, but we can use the formula to find n in terms of d:-26 = -19 + (n - 1)d-26 + 19 = (n - 1)d-7 = (n - 1)d(n - 1) = -7/dn = 1 - 7/dHowever, we can find d using another approach. Let's assume the sequence is -19, -18, -17, ..., -26, and d = 1:-26 = -19 + (n - 1)1-26 + 19 = n - 1-7 = n - 1n = 8 - 1 + 1n = 8If d = 1, then n = 8.Let's verify if d = 1 is correct:a₂ = a₁ + d = -19 + 1 = -18a₃ = a₂ + d = -18 + 1 = -17...a₈ = a₇ + d = -25 + 1 = -26Yes, d = 1 is correct. Answer: The number of terms in the arithmetic sequence is 8.
