The fifth term of an arithmetic sequence is 22 and the 15th term is 62. Find the 100th term and the sum of the first 60 terms. I'm having trouble figuring out how to do this.
T15-T5 = 10d = 40
so, d = 4
Now you can find T1 and T60
Then the sum is 30(T1+T60)
First you use the known to find the unknown....
U2=22, n=5
U15=62, n=15
You use from the formular
Un=a+(n-1)d
from the fith term
22=a+(5-1)d
22=a+4d--------equation 1
From the 15th term
62=a+(n-1)d
62=a+(15-1)d
22=a+14d-------equation 2
a+4d=22 you subtract
a+14d=62
-10d=-40
d=4
Substitute the value or d in equa 1
a+4d=22
a+4×4=22
a+16=22
a=22-16
a=6
Now use the formular to find the 100th term
Un=a+(n-1)d
Good luck
To find the nth term of an arithmetic sequence, two pieces of information are needed: the first term (a₁) and the common difference (d).
Given that the fifth term is 22, we can determine the first term (a₁) using the formula:
a₅ = a₁ + (5-1)d
Substituting the given values, we get:
22 = a₁ + 4d
Similarly, we can find the first term (a₁) using the fifteenth term:
a₁₅ = a₁ + (15-1)d
Substituting the given values and using the previous equation to eliminate a₁, we get:
62 = 22 + 14d
Now, we have a system of two equations with two variables (a₁ and d):
22 = a₁ + 4d
62 = 22 + 14d
We can solve this system to find the values of a₁ and d.
First, let's solve for d:
62 - 22 = 14d
40 = 14d
d = 40/14
d = 20/7
Now that we know the value of d, we can substitute it into one of the equations to find a₁:
22 = a₁ + 4(20/7)
22 - (80/7) = a₁
(154 - 80)/7 = a₁
(74/7) = a₁
a₁ = 74/7
a₁ = 10.571
So, the first term (a₁) is approximately 10.571, and the common difference (d) is approximately 20/7.
To find the 100th term (a₁₀₀), we can use the formula for the nth term:
a₁₀₀ = a₁ + (n-1)d
Substituting the known values, we get:
a₁₀₀ = 10.571 + (100-1) * (20/7)
a₁₀₀ = 10.571 + 99 * (20/7)
a₁₀₀ = 10.571 + 2820/7
a₁₀₀ = (10.571 * 7 + 2820)/7
a₁₀₀ = (73.997 + 2820)/7
a₁₀₀ = 2893.997/7
a₁₀₀ ≈ 413.428
Therefore, the 100th term (a₁₀₀) of the arithmetic sequence is approximately 413.428.
To find the sum of the first 60 terms (S₆₀), we can use the formula for the sum of an arithmetic sequence:
Sₙ = (n/2)(a₁ + aₙ)
Substituting the known values, we get:
S₆₀ = (60/2)(10.571 + a₁₀₀)
S₆₀ = 30(10.571 + 413.428)
S₆₀ = 30(423.999)
S₆₀ = 12,719.97
Therefore, the sum of the first 60 terms (S₆₀) is approximately 12,719.97.