If TR 0 if n 0 1 if n 1 2Tn 13Tn 2 if n 1 find 72 T3 T

If T(R) = {0 if n = 0 1 if n = 1 2T(n - 1)+3T(n - 2) if n > 1 find 7(2) T(3) T(5) T(7)

Solution

1.

T(2)
= 2T(2 - 1) + 3T(2 - 2)
= 2T(1) + 3T(0)
= 2 * 1 + 3 * 0
= 2

2.

T(3)
= 2T(3 - 1) + 3T(3 - 2)
= 2T(2) + 3T(1)
= 2 * 2 + 3 * 1
= 4 + 3
= 7

3.

T(5)
= 2T(5 - 1) + 3T(5 - 2)
= 2T(4) + 3T(3)
= 2(2T(4 - 1) + 3T(4 - 2)) + 3T(3)
= 2(2T(3) + 3T(2)) + 3 * 7
= 2(2 * 7 + 3 * 2) + 21
= 2(14 + 6) + 21
= 40 + 21
= 61

4.

T(7)
= 2T(7 - 1) + 3T(7 - 2)
= 2T(6) + 3T(5)
= 2(2T(6 - 1) + 3T(6 - 2)) + 3 * 61
= 2(2T(5) + 3T(4)) + 183
= 2(2 * 61 + 3 * 20) + 183
= 2(122 + 60) + 183
= 2 * 182 + 183
= 547