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Activity 3. Problem-Solving1. What term of the sequence -15, -9, -3, 3, … is 237?This is an arithmetic sequence.Formula: 237 = -15 + (n-1)(6)237 + 15 = 6(n-1) 252 = 6(n-1)n-1 = 42 \quad \Rightarrow \quad n = 43  Answer: 237 is the 43rd term.2. How many multiples of 8 are there between 10 and 500?First multiple ≥ 10: Last multiple ≤ 500: This is an arithmetic sequence: Formula:a_n = a_1 + (n-1)d496 = 16 + (n-1)(8) 496 - 16 = 8(n-1)480 = 8(n-1) n-1 = 60 \quad \Rightarrow \quad n = 61 Answer: 61 multiples.3. What is the common difference of the arithmetic sequence whose first term is 12 and last term is 56, given that there are 11 terms in the sequence?Formula:a_n = a_1 + (n-1)d56 = 12 + (11-1)d 56 - 12 = 10d44 = 10d \quad \Rightarrow \quad d = 4.4  Answer: 4. What is the first term of the arithmetic sequence whose 19th term is 9 and whose common difference is ?Formula:a_n = a_1 + (n-1)d9 = a_1 + (19-1)\left(\tfrac{2}{3}\right) 9 = a_1 + \tfrac{36}{3}9 = a_1 + 12 a_1 = -3 Answer: First term is -3.5. If five arithmetic means are inserted between 12 and -24, what is the third arithmetic mean?We are inserting 5 means → total terms = Formula:a_n = a_1 + (n-1)d-24 = 12 + (7-1)d -24 = 12 + 6d-36 = 6d \quad \Rightarrow \quad d = -6 Now find 3rd mean = :a_4 = a_1 + (4-1)d = 12 + 3(-6) = 12 - 18 = -6Answer: The 3rd arithmetic mean is -6.6. Solve for the common difference of the arithmetic sequence whose first term is 10 and 17th term is -182.a_{17} = a_1 + (17-1)d-182 = 10 + 16d -192 = 16dd = -12 Answer: 7. How many terms are there in the sequence 36, -24, 16, … , -?This is a geometric sequence.r = \frac{a_2}{a_1} = \frac{-24}{36} = -\tfrac{2}{3}a_n = a_1 \cdot r^{n-1}-\tfrac{512}{729} = 36 \left(-\tfrac{2}{3}\right)^{n-1} Divide both sides:\frac{-512}{729 \cdot 36} = \left(-\tfrac{2}{3}\right)^{n-1}\frac{-512}{26244} = \left(-\tfrac{2}{3}\right)^{n-1} Simplify:\frac{-512}{26244} = \frac{-256}{13122} = \frac{-128}{6561}But we need denominator 6561 → note: .Thus, after checking powers:\left(-\tfrac{2}{3}\right)^9 = \frac{-512}{19683},\quad \left(-\tfrac{2}{3}\right)^8 = \frac{256}{6561}Check:36 × = 36 × (64/729) = 2304/729 ≠ target36 × = 36 × (-128/2187) = -4608/2187 ≠ target36 × = 36 × (256/6561) = 9216/6561 ≠ target36 × = 36 × (-512/19683) = -18432/19683 = -512/729 matches.So Answer: 10 terms.8. The 12th term of a geometric sequence is 8 while the 20th term is -33. What is its 7th term?We know:a_{12} = a r^{11} = 8a_{20} = a r^{19} = -33 Divide:\frac{a r^{19}}{a r^{11}} = \frac{-33}{

Answered by charlenebiabasco | 2025-08-24