Math Quiz 8.1 Review 1 Key

HPC Sequences & Series Quiz 8.1-8.3 Review Name: Key State whether each type of sequence is arithmetic, geometric, or neither, and find an explicit formula an. 1. 17, 13, 9, 5, ... d = -4 an = 17 + (n-1)(-4) = 17 - 4n + 4 an = -4n + 21 2. 1/3, 4/5, 9/7, 16/9, ... numerator = n² denominator = 2n + 1 an = n² / (2n + 1) 3. -1, 1/4, -1/16, 1/64, ... r = -1/4 an = -1(-1/4)^(n-1) 4. 3, 18, 108, 648, ... r = 6 an = 3(6)^(n-1) 5. In an arithmetic sequence, a11 = 11 and a8 = 39. Find a17. 39-11 / 8-4 = 7 39 = a1 + (8-1)(7) 39 = a1 + 49 a1 = -10 a17 = -10 + (17-1)(7) = -10 + 16(7) = 102 6. In an geometric sequence, a4 = -12 and a6 = 96. Find a14. 96 = a1 * r^(6-1) -12 = a1 * r^(4-1) 96 = -12 * r³ -8 = r³ r = -2 -12 = a1 * (-2)³ -12 = -8a1 a1 = 3/2 a14 = 3/2 * (-2)^(14-1) = 3(2)^13 = 12288 Give a recursive definition for the following sequences. Don't forget to state the first term! The formula means NOTHING without   it! 7. 2, -4, 8, -16, ... a₁ = 2 aₖ₊₁ = -2aₖ 8. 3, 6, 12, 24, ... a₁ = 3 aₖ₊₁ = 2aₖ Find the following sums. 9. 0.25 + 0.5 + 1 + 2 + ... a₁ = 0.25, r = 2, n = 15 S₁₅ = 0.25(1 - 2¹⁵) / (1 - 2) = 8191.75 10. 2 + 6 + 10 + 14 + ... + 398 + 402 402 = 2 + (n - 1)(4) 400 = 4(n - 1) 100 = n - 1 n = 101 S₁₀₁ = 101/2(2 + 402) = 20402 11. On an 8 x 8 checker board, if you placed a kernel of corn on the first square, two kernels on the second, four on the third, eight on the fourth and so on, how many kernels would be placed in the last square if this pattern continued? How many would be placed on the whole board?   1, 2, 4, 8, ... a₁ = 1, r = 2 a₆₄ = 1 * 2⁶⁴⁻¹ = 2⁶³ At the beginning of each year, Jimmy puts $150 into his retirement account. On the average, the account earns 9% interest per year. After 40 years of work, how much money does he have in his account? a₁   = 150(1.09) = 2125.50 r = 1.09 n = 40 S₄₀ = 2125.50(1 - 1.09⁴⁰) / (1 - 1.09) = $718,169.14 Find the following sums. 13. Σ(n²) from n = 1 to 12 23,392 14. Σ(3(2/7)ⁿ) from n = 0 to infinity geometric, infinite, r = 2/7 3 / (1 - 2/7) = 3 / (5/7) = 21/5