Sequence Extension 682F5E

1. **Problem Statement:** We are given two sequences and asked to extend each with three more terms. 2. **Sequence j:** $22, -16, -54, -92, ...$ 3. **Step 1: Identify the pattern in sequence j.** Calculate the differences between consecutive terms: $$-16 - 22 = -38$$ $$-54 - (-16) = -38$$ $$-92 - (-54) = -38$$ The common difference is $-38$, so this is an arithmetic sequence. 4. **Step 2: Use the arithmetic sequence formula:** $$a_n = a_1 + (n-1)d$$ where $a_1 = 22$ and $d = -38$. 5. **Step 3: Find the next three terms:** $$a_5 = 22 + 4 \times (-38) = 22 - 152 = -130$$ $$a_6 = 22 + 5 \times (-38) = 22 - 190 = -168$$ $$a_7 = 22 + 6 \times (-38) = 22 - 228 = -206$$ 6. **Sequence j extended:** $$22, -16, -54, -92, -130, -168, -206$$ 7. **Sequence b:** $-17, -102, -612, -3672, ...$ 8. **Step 1: Identify the pattern in sequence b.** Calculate the ratios between consecutive terms: $$\frac{-102}{-17} = 6$$ $$\frac{-612}{-102} = 6$$ $$\frac{-3672}{-612} = 6$$ The common ratio is $6$, so this is a geometric sequence. 9. **Step 2: Use the geometric sequence formula:** $$a_n = a_1 \times r^{n-1}$$ where $a_1 = -17$ and $r = 6$. 10. **Step 3: Find the next three terms:** $$a_5 = -17 \times 6^4 = -17 \times 1296 = -22032$$ $$a_6 = -17 \times 6^5 = -17 \times 7776 = -132192$$ $$a_7 = -17 \times 6^6 = -17 \times 46656 = -792,552$$ 11. **Sequence b extended:** $$-17, -102, -612, -3672, -22032, -132192, -792552$$