C. Find the nth term or the general term that describes each of the following sequences. 1. -4, -8, -12, ... 2. 1/2, 1/4, 1/8, ... 3. -2, 1, 7, 16
Answer (1)
To find the nth term or general term of a sequence, we need to identify a pattern or rule that can be applied to any term number n in the sequence.
For the sequence -4, -8, -12, ...:
This is an arithmetic sequence because each term is obtained by subtracting the same amount from the previous term.
The first term, a, is -4. The common difference, d, can be found by subtracting the first term from the second term: -8 - (-4) = -4.
The nth term of an arithmetic sequence can be found using the formula: a n = a + ( n − 1 ) ⋅ d Plugging in the values, we get: a n = − 4 + ( n − 1 ) ( − 4 ) = − 4 − 4 n + 4 = − 4 n
So, the nth term for the sequence is − 4 n .
For the sequence 2 1 , 4 1 , 8 1 , ... :
This is a geometric sequence where each term is obtained by multiplying the previous term by the common ratio.
The first term, a, is 2 1 , and the common ratio, r, can be found by dividing the second term by the first term: 4 1 ÷ 2 1 = 2 1 .
The nth term of a geometric sequence can be found using the formula: a n = a ⋅ r ( n − 1 ) Plugging in the values, we get: a n = 2 1 ⋅ ( 2 1 ) ( n − 1 ) = 2 n 1
So, the nth term for this sequence is 2 n 1 .
For the sequence -2, 1, 7, 16, ...:
This sequence does not follow a simple arithmetic or geometric pattern. However, it seems to follow a quadratic pattern. To find such a pattern, we can assume a general quadratic formula for the nth term: a n = a n 2 + bn + c
Using the first three terms to create a system of equations:
When n = 1 , − 2 = a ( 1 ) 2 + b ( 1 ) + c
When n = 2 , 1 = a ( 2 ) 2 + b ( 2 ) + c
When n = 3 , 7 = a ( 3 ) 2 + b ( 3 ) + c
Solving this system of equations, we find that: a = 2 3 , b = − 2 3 , c = 0
So, the nth term for this sequence is a n = 2 3 n 2 − 2 3 n .
