1. Write the first terms of each of the following sequences whose n th term are 

(i) a_n = 3n + 2
(ii) a_n = (n - 2)/3
(iii) a_n = 3ⁿ
(iv) a_n = (3n - 2)/5
(v) a_n = (-1)ⁿ2ⁿ
(vi) a_n = n(n - 2)/2
(vii) a_n = n² - n + 1
(viii) a_n = n² - n + 1
(ix) a_n = (2n - 3)/6

Solution

(i) a_n = 3n + 2
Let n = 1, 2, 3, 4, 5, then
First five terms, a1 = 3×1 + 2 = 3+2 = 5
a₂ = 3×2 + 2 = 6+2 = 8
a₃ = 3×3 + 2 = 9+2 = 11
a₄ = 3×4 + 2 = 12+2 = 14
a₅ = 3×5 + 2 = 15+2 = 17

(iii) a_n = 3ⁿ
Let n = 1, 2, 3, 4, 5, then
a₁ = 3¹ = 3
a₂ = 3² = 3×3 = 9
a₃ = 3³ = 3×3×3 = 27
a₄ = 3⁴ = 3×3×3×3 = 81
a₅ = 3⁵ = 3×3×3×3×3 = 243

2. Find the indicated terms in each of the following sequences whose nth terms are:
(i) a_n = 5n - 4; a₁₂ and a₁₅
(ii) a_n = (3n - 2)/(4n + 5); a₇ and a₈
(iii) a_n = n(n-1)(n-2); a₅ and a₈
(iv) a_n = (n - 1)(2 - n)(3 + n); a₁₁ a₂₁ a₃
(v) a_n = (-1)ⁿ n; a₃ , a₅ , a₈

Solution

3. Find the next five terms of each of the following sequences given by: 
(i) a₁ = 1, a_n = a_{n - 1} + 2, n ≥ 2 
(ii) a₁ = a₂ = 2 , a_n = a_{n - 1} - 3, n > 2 
(iii) a₁ = -1, a_n = (a_{n - 1} )/n , n ≥ 2 
(iv) a₁ = 4, a_n = 4 a_{n - 1} + 3, n > 1

Solution