Math Sequences and Series Key

Algebra 2 Honors Final Review Sequences & Series Find a formula for the nth term, of each sequence 18, 11, 4, -3, ... tₙ = 18 + (n-1)(-7) tₙ = -7n + 25 16, 24, 36, 54, ... r = 3/2 tₙ = 16(3/2)^(n-1) Given the arithmetic sequence 9, 4, -1, ... find each of the following. The value of t₁₇ t₁₇ = 9 + (17-1)(-5) t₁₇ = -71 Which term is -231? -231 = -5n + 14 n = 49 Find S₃₈ S₃₈ = 38(9 + (-126))/2 S₃₈ = -3173 Write a rule for the nth term of the arithmetic sequence. d = 7, t₀ = 54 54 = t₁ + (7)(7) t₁ = 5 tₙ = 5 + (n-1)(7) tₙ = 7n - 2 t₄ = 27, t₁₁ = 69 d = (69-27)/(11-4) = 6 27 = t₁ + 3(6) t₁ = 9 tₙ = 9 + (n-1)(6) tₙ = 6n + 3 8. Insert three arithmetic means between 6 and 26. 26 = 6 + (4)d d = 5 6, 11, 16, 21, 26 Write a rule for the nth term of the geometric sequence. r = 5, t₇ = 200 200 = t₁(5)⁴ t₁ = 8/625 tₙ = 8/625(5)^(n-1) t₃ = 16, t₅ = 16/9 16/9 = t₁(r)² t₁ = 144 tₙ = 144(1/3)^(n-1) or tₙ = 144(-1/3)^(n-1) Find t₁₀ of the geometric sequence 1, -2, 4, -8, 16, ... t₁₀ = 1(-2)⁹ t₁₀ = -512 12. The temperature of water in a kettle is 50°F when it is placed on the stove. Its temperature n seconds after being placed on the stove is 7% more than 1 second earlier. Find the temperature of the water a) 20 seconds and b) 1 minute after it is placed on the stove. 50(1.07) = 53.5 a) t₂₀ = 53.5(1.07)^19 ≈ 193.48° b) t₆₀ = 53.5(1.07)^59 ≈ 2897.32° 13. Find the sum of the first 20 terms of the geometric series 3/4 + 3/2 + 3 + 6 + ... S_20 = (3/4)(1 - 2^20) / (1 - 2) = 3145725/4 14. Find the geometric mean of 1/12 and 1/18. 1/12, x, 1/18 x^2 = (1/12)(1/18) x = √(1/216) = 1/3√6 15. Insert 4 geometric means between -4 and -972. -4, r, r^2, r^3, r^4, -972 -972 = -4(r)^5 r = 3 -4, -12, -36, -108, -324, -972 Evaluate. You must use a formula when appropriate. 16. Σ_(n=4)^(500) (2n - 1) t_4 = 2(4) - 1 = 7 t_500 = 2(500) - 1 = 999 S_497 = (497)(7 + 999)/2 = 249,991 17. Σ_(k=3)^(10) 6(3)^(k-2) t_3 = 6(3)^(3-2) = 18 t_10 = 6(3)^(10-2) = 59049 S_8 = (18(1 - 3^8))/(1 - 3) = 59040 18. Σ_(k=1)^(4) (2/3)^(k-1) t_1 = 2/3 t_2 = 4/9 t_3 = 8/27 t_4 = 16/81 S_4 = 110/81 19. Find the sum of the series 24 - 9.6 + 0.96 - 0.192 + ... r = -1/5 S = 24 / (1 - (-1/5)) = 20 Write in sigma notation. 20. 1/3 + 2/5 + 3/7 + 4/9 + ... Σ_(n=1)^(∞) n/(2n+1) 21. 2 + 2*3 + 2*3² + ... + 2*3¹⁶ Σ_(n=1)^(17) 2(3)^(n-1) Find the sum of each series. Note, for arithmetic and geometric series, you may use the formula. You must use other methods to find non-arithmetic and non-geometric sums.   22. Odd integers from 100-350.   n = 125 S_125 = (125)(101 + 349)/2 = 28,125 23. The positive four-digit integers divisible by 12. 1008 + 1020 + ... + 9996 9996 = 1008 + (n-1)12 n = 750 S_750 = (750)(1008 + 9996)/2 = 4,126,500 Use Arithmetic and Geometric Sequences and Series to solve the following applications. A small hardware store makes a profit of $10,000 during its first year. The store owner sets a goal of increasing profits by $700 each year for 7 years. Assuming that this goal is met, find the total profit during the first 8 years of business.   t₈ = 10000 + (8-1)700 = 14900 S₈ = 8(10000 + 14900)/2 = $99,600 A bouncy ball is dropped from the top of a 200 foot building. It rebounds to 93% of its original height after each successive bounce. After bouncing and rebounding 15 times, how far has the ball traveled?   S_down = 200(1 - 0.93^15) / (1 - 0.93) S_up = 186(1 - 0.93^15) / (1 - 0.93) Total distance = S_down + S_up ≈ 3657.62 ft Write each of the following as a fraction. 0.7 7/9 5.15 170/33 Write the first five terms of the sequence. t₁ = 8, tₙ = 5 * tₙ₋₁ 8, 40, 200, 1000, 5000 t₁ = 4, tₙ = (tₙ₋₁ + 9) 4, 13, 22, 31, 40 t₁ = 2, tₙ = n * tₙ₋₁ 2, 4, 12, 48, 240 t₁ = 4, tₙ = (tₙ₋₁)ⁿ 4, 16, 4096, 4096⁴, 4096⁴⁰⁹⁶ Expand (m² + 2n)⁴ (m²)⁴ + 4(m²)³(2n)¹ + 6(m²)²(2n)² + 4(m²)(2n)³ + (2n)⁴ = m⁸ + 8m⁶n + 24m⁴n² + 32m²n³ + 16n⁴ Find the 5th term of (a - 3b)⁷ 7C₄(a)³(-3b)⁴ = 2835a³b⁴ Find the 4th term of (3e + 2f)⁷ 7C₃(3e)⁴(2f)³ = 22,680e⁴f³ Find the term containing b⁵ in (a - b)¹⁶ 16C₅(a)¹¹(-b)⁵ = -4368a¹¹b⁵