Determine whether the sequence is arithmetic. If so, find the common difference. (Assume that n begins with 1.)a n=2(1/5)n
Oh, look at that fancy equation trying to be all mathematical! Well, let me put on my arithmetic glasses and see if I can figure it out.
The formula you've given me is n = 2(1/5)n. Now, let's simplify this a bit.
If we distribute the 2 into the parentheses, we get n = 2/5n.
Hm, well, it seems like the common difference is 2/5. So, yes, it is an arithmetic sequence. But let me tell you, it's a pretty wacky and fractional one! Keep an eye out for those naughty fractions, they always like to stir things up.
To determine if the sequence is arithmetic and find the common difference, we need to analyze the formula given and see if it follows the characteristics of an arithmetic sequence.
The formula for the nth term, an, is given as:
an = 2(1/5)n
To determine if this formula represents an arithmetic sequence, we need to check if the terms have a constant difference.
Let's calculate the first few terms to see if they follow a pattern:
When n = 1:
a1 = 2(1/5)^1 = 2/5
When n = 2:
a2 = 2(1/5)^2 = 2/25
When n = 3:
a3 = 2(1/5)^3 = 2/125
From the calculated terms, we can see that the terms are not following a constant difference. Hence, the sequence represented by the formula is not arithmetic.
Therefore, there is no common difference to find in this case.
To determine whether the given sequence is arithmetic, we need to check whether the difference between consecutive terms is constant. The formula for an arithmetic sequence is generally given by:
aₙ = a₁ + (n - 1) * d
In this case, we are given the formula for the nth term of the sequence:
aₙ = 2 * (1/5) * n
To find the common difference (d), we will substitute n = 2 into the formula and subtract the result from n = 1 for the given sequence:
a₂ = 2 * (1/5) * 2 = 4/5
a₁ = 2 * (1/5) * 1 = 2/5
Now, we can calculate the difference between a₂ and a₁:
d = a₂ - a₁ = (4/5) - (2/5) = 2/5
Therefore, the common difference (d) for the given sequence is 2/5. Since the difference between consecutive terms is constant, we can conclude that the sequence is indeed arithmetic.