I swear, last question!
Calculate S(31) for the arithmetic sequence in which a(9) = 17 and the common difference is d = -2.1
-46
-29.2
32.7
71.3
Usually, I can figure these out but I'm stuck:(
a(n) = a(1) + (n-1)d
a(9) = a(1) + (9-1)d
You have a(9), you have d
Use those to get a(1) from the above mentioned.
Then, plug those values into the sum formula I mentioned on the other question.
Ok, I'm so sorry, but I got so lost.
For some reason, (I think) I'm getting it wrong at determining a(1).
I don't know what the heck I'm doing, but I've gotten either 5.9, -5.9, or 11.1 for a(1) and none of them work for the other formula. I feel super dumb right now, is there any chance you could show me again how to get a(1)? Sorry
Wait-- I made a dumb typo. I think it's 71.3 (oh my gosh I'm dying)
a(9) = a(1) + (n-1)d
a(9) = 17, d = -2.1
17 = a(1) + (9-1)*-2.1
a(1) = 17 + 16.8
= 33.8
S(31) = (n/2)[2a(1) + (n-1)d]
= (31/2)[2(33.8) + 30(-2.1)]
= (31/2)(67.6 - 63)
= (31/2)(4.6)
= 31*2.3
= 71.3
Yes, you're correct
No worries! I'm here to help you understand how to solve this problem.
To find the sum of an arithmetic sequence, you can use the formula:
S(n) = (n/2)(a(1) + a(n))
Where:
- S(n) represents the sum of the sequence up to the nth term.
- n is the number of terms in the sequence.
- a(1) is the first term of the sequence.
- a(n) is the nth term of the sequence.
In your case, you want to calculate S(31) for the arithmetic sequence with a(9) = 17 and a common difference of -2.1. Since you're given the 9th term, you can use it to find the first term, a(1).
To find a(1), you can use the formula:
a(n) = a(1) + (n - 1)d
Plugging in the values you have:
17 = a(1) + (9 - 1)(-2.1)
Now you can solve for a(1):
17 = a(1) + 8(-2.1)
17 = a(1) - 16.8
a(1) = 17 + 16.8
a(1) = 33.8
Now that you have a(1) and a(n), you can calculate S(31) using the formula:
S(n) = (n/2)(a(1) + a(n))
Plugging in the values:
S(31) = (31/2)(33.8 + a(31))
You need to find a(31) to solve the equation. Using the same arithmetic progression formula, you can find it:
a(n) = a(1) + (n - 1)d
a(31) = 33.8 + (31 - 1)(-2.1)
a(31) = 33.8 + 30(-2.1)
Now you can calculate a(31):
a(31) = 33.8 - 63
a(31) = -29.2
Now that you have a(31), you can calculate S(31):
S(31) = (31/2)(33.8 + (-29.2))
S(31) = (31/2)(4.6)
Now it's just a matter of calculating the expression:
S(31) = (31/2) * 4.6
S(31) = 71.3
Therefore, the correct answer is 71.3.