Consider the sequence: 3^2x, 2×3^2x, 3^(2x+1), 4×3^(2x), 5×3^(2x),... Is the given sequence an AP? If it is an AP, find the 28th term
To determine whether the sequence is an arithmetic progression (AP), we need to look at the differences between consecutive terms.
The differences are:
2×3^2x - 3^2x = 2×(3^2x) - (3^2x) = (2-1)×(3^2x) = 3^2x
3^(2x+1) - 2×3^2x = 3^(2x)×3 - 2×3^(2x) = (3-2)×(3^2x) = 3^2x
4×3^(2x) - 3^(2x+1) = 4×(3^2x) - (3^(2x)×3) = (4-3)×(3^2x) = 3^2x
5×3^(2x) - 4×3^(2x) = 5×(3^2x) - 4×(3^2x) = (5-4)×(3^2x) = 3^2x
From the differences, we can see that the common difference between consecutive terms is 3^2x. Therefore, the given sequence is an AP.
To find the 28th term, we need to determine the general term of the AP. Since the common difference is 3^2x, the general term can be written as:
a(n) = a(1) + (n-1)d
where a(1) represents the first term, d represents the common difference, and n represents the position of the term we want to find.
In this case, we are given that the first term is 3^2x. Therefore, a(1) = 3^2x.
Substituting the values into the general term formula, we have:
a(28) = 3^2x + (28-1)(3^2x)
= 3^2x + 27(3^2x)
= 3^2x + 3^3x
= 3^2x(1 + 3)
= 3^2x(4)
Therefore, the 28th term of the given sequence is 4×(3^2x).
To determine if the given sequence is an Arithmetic Progression (AP), we need to check if the difference between consecutive terms is constant.
Let's calculate the differences between consecutive terms:
Difference between 2nd and 1st term:
2×3^2x - 3^2x = 3^2x(2 - 1) = 3^2x
Difference between 3rd and 2nd term:
3^(2x+1) - 2×3^2x = 3^2x(3^1 - 2) = 3^2x
Difference between 4th and 3rd term:
4×3^(2x) - 3^(2x+1) = 3^2x(4 - 3) = 3^2x
As we can see, the difference between consecutive terms is constant, equal to 3^2x. Therefore, the given sequence is an Arithmetic Progression.
To find the 28th term, we can use the formula for the nth term of an Arithmetic Progression:
nth term = first term + (n-1) * common difference
In this case, the first term is 3^2x (since it was not explicitly given), and the common difference is 3^2x.
To find the 28th term, we substitute n = 28 into the formula:
28th term = 3^2x + (28-1) * 3^2x
= 3^2x + 27 * 3^2x
= 3^2x(1 + 27)
= 3^2x(28)
= 28 * 3^2x
Therefore, the 28th term of the given sequence is 28 * 3^2x.