THE General Term

1.

Let T(1) = 7, T(2) = 13, T(3) = 25, T(4) = 49:
Then
T(1) = 7
T(2) = T(1) × 2 - 1 = 7 × 2 - 1 = 13
T(3) = T(2) × 2 - 1 = 13 × 2 - 1 = 25
T(4) = T(3) × 2 - 1 = 25 × 2 - 1 = 49

∴The next two terms are:
T(5) = T(4) × 2 - 1 = 49 × 2 - 1 = 97
T(6) = T(5) × 2 - 1 = 97 × 2 - 1 = 193


2.

Let T(n) be the general term:
Then T(n) = T(n - 1) × 2 - 1
　　　　　= (T(n - 2) × 2 - 1) × 2 - 1
　　　　　= T(n - 2) × 2² - 2 - 1
　　　　　= (T(n - 3) × 2 - 1) × 2² - 2 - 1
　　　　　= T(n - 3) × 2³ - 2² - 2 - 1
　　　　　= (T(n - 4) × 2 - 1) × 2³ - 2² - 2 - 1
　　　　　= T(n - 4) × 2^4 - 2³ - 2² - 2 - 1
　　　　　=
　　　　　：
　　　　　：
　　　　　：
　　　　　= T(1) × 2^(n - 1) + (- 2^(n - 2) - 2^(n - 3) - 2^(n - 4) - … - 2³ - 2² - 2 - 1)
　　　　　= 7 × 2^(n - 1) - (1 + 2 + 2² + … + 2^(n - 2))
　　　　　= 7 × 2^(n - 1) - 1 × (2^(n - 1) - 1)/(2 - 1)
　　　　　= 7 × 2^(n - 1) - 2^(n - 1) + 1
　　　　　= 6 × 2^(n - 1) + 1

∴The general term is T(n) = 6 × 2^(n - 1) + 1

1. The next two terms are
97, 193

2. Let an be the general term for n
a1 = 7 = 7 x 2 - 1
a2 = 13 = 13 x 2 - 1
a3 = 25 = 25 x 2 - 1
... ...
an = 2n - 1
Therefore, the general term is 2n - 1