2.6, −5.2, 10.4, −20.8, . . .
6th term?
what is an easier way to find this?
To find the 6th term in the given sequence, let's first analyze the pattern. It appears that each term in the sequence is obtained by multiplying the previous term by -2. Therefore, the pattern can be written as:
Term 1: 2.6
Term 2: 2.6 * -2 = -5.2
Term 3: -5.2 * -2 = 10.4
Term 4: 10.4 * -2 = -20.8
Term 5: -20.8 * -2 = 41.6
From here, we can observe that each term alternates in sign (positive, negative, positive, and so on) while doubling in magnitude. Therefore, for the 6th term, we can perform the same operation:
Term 6: 41.6 * -2 = -83.2
Thus, the 6th term in the sequence is -83.2.
As for an easier way to find the 6th term, you can use the formula for the nth term of a geometric sequence:
an = a1 * r^(n-1)
Where:
- an represents the nth term
- a1 represents the first term
- r represents the common ratio
In our case:
- First term (a1): 2.6
- Common ratio (r): -2
- n: 6
Plugging in these values into the formula:
a6 = 2.6 * (-2)^(6-1)
= 2.6 * (-2)^5
= 2.6 * (-32)
= -83.2
Following this method, we arrive at the same result: the 6th term is -83.2.