Theory question 109
Sequences and Series(a) The nth term of a sequence is given by Tn = 4Tn−1 − 3. If twice the third term is five times the second term, find the first three terms of the sequence.
(b) Given that the matrix (2 0 1 / 5 −3 1 / 0 4 6) multiplied by the column (1 / m / r) equals the column (k / 2 / 26), find the values of constants k, m and r. (Matrices are written row by row.)
Show the model answer
Model answer
(a) T2 = 4T1 − 3, T3 = 4T2 − 3 = 16T1 − 15
2T3 = 5T2: 32T1 − 30 = 20T1 − 15, T1 = 5/4
First three terms: 1¼, 2, 5
(b) 2 + r = k; 5 − 3m + r = 2; 4m + 6r = 26
From the second, r = 3m − 3; then 4m + 18m − 18 = 26, m = 2
r = 3, k = 5
Explanation
Express T₂ and T₃ in terms of T₁; multiply out the matrix row by row.