Answer: Using n = 1, 2, 3, 4, 5, the first five terms can be calculated. If n = 1, a1 = (1)2 – 1 + 1 a1 = 1 – 1 + 1 a1 = 1 If n = 2, a2 = (2)2 – 2 + 1 a2 = 4 – 2 + 1 a2 = 3 If n = 3, a3 = (3)2 – 3 +...
A sequence is defined by an = n3 – 6n2 + 11n – 6, n ∈ N. Show that the first three terms of the sequence are zero and all other terms are positive.
Answer: Using n = 1, 2, 3, the first three terms can be calculated. If n = 1, a1 = (1)3 – 6(1)2 + 11(1) – 6 a1 = 1 – 6 + 11 – 6 a1 = 12 – 12 a1 = 0 If n = 2, a2 = (2)3 – 6(2)2 + 11(2) – 6 a2 = 8 –...
Find the first four terms of the sequence defined by a1 = 3 and an = 3an–1 + 2, for all n > 1.
Answer: Using n = 1, 2, 3, 4, the first four terms can be calculated. If n = 1, a1 = 3 If n = 2, a2 = 3a2–1 + 2 a2 = 3a1 + 2 a2 = 3(3) + 2 a2 = 9 + 2 a2 = 11 If n = 3, a3 = 3a3–1 + 2 a3 = 3a2 + 2...
Write the first five terms in each of the following sequences: (i) a1 = 1, an = an–1 + 2, n > 1 (ii) a1 = 1 = a2, an = an–1 + an–2, n > 2
Answer: (i) Using n = 1, 2, 3, 4, 5, the first five terms can be calculated. If n = 1, a1 = 1 If n = 2, a2 = a2–1 + 2 a2 = a1 + 2 a2 = 1 + 2 a2 = 3 If n = 3, a3 = a3–1 + 2 a3 = a2 + 2 a3 = 3 + 2...
Write the first five terms in each of the following sequence: a1 = a2 =2, an = an–1 – 1, n > 2
Answer: Using n = 1, 2, 3, 4, 5, the first five terms can be calculated. If n = 1, a1 = 2 If n = 2, a2 = 2 If n = 3, a3 = a3–1 – 1 = a2 – 1 = 2 – 1 = 1 If n = 4, a4 = a4–1 – 1 = a3 – 1 = 1 –...
The Fibonacci sequence is defined by a1 = 1 = a2, an = an–1 + an–2 for n > 2. Find (an+1)/an for n = 1, 2, 3, 4, 5.
3Answer: an = an–1 + an–2 If n = 1, (an+1)/an = (a1+1)/a1 = a2/a1 = 1/1 = 1 a3 = a3–1 + a3–2 = a2 + a1 = 1 + 1 = 2 If n = 2, (an+1)/an = (a2+1)/a2 = a3/a2 = 2/1 = 2 a4 = a4–1 + a4–2 = a3 + a2 = 2 +...