The 8th term in an arithmetic sequence is 5, and the sum of the first 10 terms is 20. Find the common difference and the first term of the sequence.
Alright so in an arithmetic squence the difference d between any two successive numbers is constant. So if the first term is x, then the second term is (x+d), and the third term is (x+2d), etc. up to the tenth term in this case which is (x+9d). We know that the eighth term is 5 so this gives...
x+7d=5
and we know that the sum of the first ten terms is 20 which gives...
x+(x+d)+(x+2d)+(x+3d)+...+(x+9d)=20
or..
10x+45d=20
Solve these two equations for x and d..
x+7d=5
10x+45d=20
I get... d=1.2 and x=-3.4
Hope that helps
!
Well, I guess you could say that the common difference is feeling a little lost, with a value of 1.2. And as for the first term, it's so negative, it's practically chilling at a temperature of -3.4 degrees. Hopefully, it warms up a bit in the sequence!
To find the common difference and the first term of the sequence, we can solve the system of equations:
x + 7d = 5 ...(1)
10x + 45d = 20 ...(2)
Let's solve for x and d:
First, we can rewrite equation (1) as:
x = 5 - 7d
Substitute this value of x into equation (2):
10(5 - 7d) + 45d = 20
50 - 70d + 45d = 20
-25d = -30
d = -30 / -25
d = 6/5 or 1.2
Now plug this value of d back into equation (1) to find x:
x + 7(1.2) = 5
x + 8.4 = 5
x = 5 - 8.4
x = -3.4
Therefore, the common difference (d) of the sequence is 1.2 and the first term (x) is -3.4.
To solve this problem, you correctly set up two equations using the information given. From there, you can solve for the common difference (d) and the first term (x).
Let's solve the first equation: x + 7d = 5
To isolate x, subtract 7d from both sides:
x = 5 - 7d
Now substitute this expression for x in the second equation: 10x + 45d = 20.
Replace x with 5 - 7d:
10(5 - 7d) + 45d = 20
Distribute the 10:
50 - 70d + 45d = 20
Combine like terms:
50 - 25d = 20
Subtract 50 from both sides:
-25d = -30
Divide both sides by -25 to solve for d:
d = -30 / -25
Simplifying, we get:
d = 1.2
Now, substitute this value of d back into the first equation to solve for x:
x + 7d = 5
x + 7(1.2) = 5
x + 8.4 = 5
Subtract 8.4 from both sides:
x = 5 - 8.4
Simplifying:
x = -3.4
So, the common difference is 1.2 and the first term is -3.4.