What are the first and last term of an arithmetic sequence when its arithmetic means 36.6, 33.2, 29.8, and 26.4
To find the first and last term of an arithmetic sequence, we need to know the common difference and at least two terms of the sequence. Given the arithmetic means, we can use the formula for the nth term of an arithmetic sequence to solve for the common difference.
The formula for the nth term of an arithmetic sequence is:
an = a1 + (n-1)d
Where:
an = nth term of the sequence
a1 = first term of the sequence
n = position of the term in the sequence
d = common difference between the terms
Given the arithmetic means provided, we can find the common difference by subtracting each term from the next term.
The common difference d is calculated as follows:
d = 33.2 - 36.6 = -3.4
d = 29.8 - 33.2 = -3.4
d = 26.4 - 29.8 = -3.4
Since the common difference is -3.4, we can now find the first and last terms of the arithmetic sequence.
To find the first term a1, we can use the formula:
a1 = an - (n-1)d
Using the values provided, we can substitute an = 36.6, n = 1, and d = -3.4 into the formula:
a1 = 36.6 - (1-1)(-3.4)
a1 = 36.6 - 0
Therefore, the first term of the sequence is 36.6.
To find the last term of the sequence, we need to determine the position of the last term. Given that we have already found the common difference to be -3.4, we can use the formula for the nth term of the arithmetic sequence to find the position.
The formula to find the position is:
n = (an - a1)/d + 1
Substituting the values into the formula:
n = (26.4 - 36.6)/(-3.4) + 1
n = -10.2/(-3.4) + 1
n = 3 + 1
Therefore, the last term is the term in position 4 of the sequence.
Finally, we can find the last term using the formula:
an = a1 + (n-1)d
Substituting the values into the formula:
an = 36.6 + (4-1)(-3.4)
an = 36.6 - 3(3.4)
an = 36.6 - 10.2
Therefore, the last term of the sequence is 26.4.