Since each term in the sequence below is equal to the sum of the two previous terms, d = b + c; e = c + d, since c and d are the two terms previous to e. If e = c + d, then, by subtracting c from both sides of the equation, d = e – c. In the same way, f = d + e, the terms that precede it, and that equation can be rewritten as d = f – e; d = b + c, and c = a + b. Therefore, d = b + (a + b), d = a + 2b. However, d is not equal to e – 2b; d = e – c, and c = a + b, not 2b, since a is not equal to b.

Since the rule of the sequence is each term is 16 more than –4 times the previous term, to find the value of y, subtract 16 from –80 and divide by –4:\(\frac{-80-16}{-4}=-\frac{-96}{-4}=24\). In the same way, the value of x is\(\frac{(24-16)}{-4}=\frac{8}{-4}=-2\). Therefore, the value of x + y = –2 + 24 = 22.

Continue the sequence; 28.5 is the fourth term of the sequence. The fifth term is \(\left ( \frac{28.5}{2} \right )\) – 2 = 14.25 – 2 = 12.25. The sixth term is \(\left ( \frac{12.25}{2} \right )\) – 2 = 6.125 – 2 = 4.125, the seventh term is \(\left ( \frac{4.125}{2} \right )\) – 2 = 2.0625 – 2 = 0.0625. Half of this number minus two will yield a negative value, so the eighth term of the sequence is the first term of the sequence that is a negative number.

Since the rule of the sequence is each term is 20 less than five times the previous term, to find the value of x, add 20 to 0 and divide by 5:\(\frac{(0+20)}{5}=\frac{20}{5}\) = 4. In the same way, the value of y is \(\frac{(-120+20)}{5}=\frac{-100}{5}=-20\). Therefore, the value of x + y = 4 + –20 = –16.

Since the rule of the sequence is each term is nine more than ¹⁄₃ the previous term, to find the value of x, multiply the last term, 36, by ¹⁄₃, then add 9: (36)(¹⁄₃) = 12, 12 + 9 = 21. In the same way, the value of y is 21(¹⁄₃) + 9 = 7 + 9 = 16. Therefore, the value of y – x = 16 – 21 = –5.

Since the rule of the sequence is each term is two less than three times the previous term, multiply the last term, –53, by 3, then subtract 2: (–53)(3) = –159 – 2 = –161.