B. Find the nth term (general term) of the following sequences. 1) 3, 5, 7, 9, 11, ... 2) 7, 11, 15, 19, 23, ... 3) 2, 8, 18, 32, ... 4) -1, 1, 3, 5, 7, 9, ... 5) -2, 1, 4, 7, 10, ...
Answer (1)
Let's find the nth term (general term) for each of the given sequences.
Sequence: 3, 5, 7, 9, 11, ...
This sequence is an arithmetic sequence where each term increases by 2. The first term a 1 is 3 and the common difference d is 2.
The general term a n for an arithmetic sequence can be found using the formula:
a n = a 1 + ( n − 1 ) d
Substituting the values, we get:
a n = 3 + ( n − 1 ) × 2 = 3 + 2 n − 2 = 2 n + 1
So, the nth term is 2 n + 1 .
Sequence: 7, 11, 15, 19, 23, ...
This is also an arithmetic sequence where each term increases by 4. Here, a 1 = 7 and d = 4 .
Using the formula for the nth term:
a n = a 1 + ( n − 1 ) d
a n = 7 + ( n − 1 ) × 4 = 7 + 4 n − 4 = 4 n + 3
So, the nth term is 4 n + 3 .
Sequence: 2, 8, 18, 32, ...
This sequence appears to follow a quadratic pattern. We can use a difference method to find the underlying pattern.
First differences: 6 , 10 , 14 , … Second differences: 4 , 4 , …
Since the second differences are constant, it's a quadratic sequence, and a n = a n 2 + bn + c .
Solving, we get: a = 2 , b = 0 , c = 0
a n = 2 n 2
So, the nth term is 2 n 2 .
Sequence: -1, 1, 3, 5, 7, 9, ...
This is another arithmetic sequence where each term increases by 2. Here, a 1 = − 1 and d = 2 .
Using the nth term formula:
a n = a 1 + ( n − 1 ) d
a n = − 1 + ( n − 1 ) × 2 = − 1 + 2 n − 2 = 2 n − 3
So, the nth term is 2 n − 3 .
Sequence: -2, 1, 4, 7, 10, ...
This sequence is arithmetic with a common difference of 3. Here, a 1 = − 2 and d = 3 .
Using the nth term formula:
a n = a 1 + ( n − 1 ) d
a n = − 2 + ( n − 1 ) × 3 = − 2 + 3 n − 3 = 3 n − 5
So, the nth term is 3 n − 5 .
