the second term of an arithmetic sequence is 15,mand the sixth term is 43. what is the 289th term?
a+d = 15
a+5d = 43
gives
a=8
d=7
a+288d = 8 + 2016 = 2024
To find the 289th term of an arithmetic sequence, we first need to find the common difference (d).
The formula to find the nth term of an arithmetic sequence is:
Term_n = first term + (n - 1) * common difference
Given that the second term is 15, we can substitute this into the formula:
15 = first term + (2 - 1) * d
15 = first term + d
Similarly, we can use the sixth term to find the common difference:
43 = first term + (6 - 1) * d
43 = first term + 5d
We have two equations:
15 = first term + d
43 = first term + 5d
Subtracting the first equation from the second equation eliminates the first term:
43 - 15 = first term + 5d - (first term + d)
28 = 4d
Dividing both sides of the equation by 4 gives us the value of the common difference:
d = 28/4
d = 7
Now that we know the value of the common difference (d), we can find the 289th term:
Term_289 = first term + (289 - 1) * common difference
Term_289 = first term + 288 * 7
Therefore, to find the 289th term of the arithmetic sequence, we need to know the value of the first term.
To find the 289th term of an arithmetic sequence, we need to determine the common difference (d) first.
In an arithmetic sequence, the common difference (d) is the constant value added or subtracted to each term to obtain the next term.
We are given the second term (a2) as 15, and the sixth term (a6) as 43.
Using the formula to find the nth term of an arithmetic sequence:
an = a1 + (n - 1) * d
Where:
an is the nth term
a1 is the first term
n is the position of the term in the sequence
d is the common difference
We can substitute the given values into the formula to solve for the common difference (d).
a2 = a1 + (2 - 1) * d
15 = a1 + d
a6 = a1 + (6 - 1) * d
43 = a1 + 5d
Now, we have a system of equations:
a1 + d = 15 ...(1)
a1 + 5d = 43 ...(2)
Subtracting equation (1) from equation (2), we can eliminate a1:
a1 + 5d - (a1 + d) = 43 - 15
4d = 28
d = 7
Now that we have the common difference (d), we can find the first term (a1) by substituting it into equation (1) or (2):
a1 = 15 - d
a1 = 15 - 7
a1 = 8
Finally, to find the 289th term (a289), we can use the formula:
a289 = a1 + (289 - 1) * d
a289 = 8 + 288 * 7
a289 = 8 + 2016
a289 = 2024
Therefore, the 289th term of the arithmetic sequence is 2024.