please explain #urgent
the sum to infinity of a convergent series is 243. the sum of the first five terms is 242. how do you determine the values of the common ratio and the first term
Use the formulas for both "sum of terms" and "sum of infinite number of terms"
Sum to infinity = a/(1-r)
a/(1-r) = 243 , ---- #1
( I recognized 243 as 3^5)
a = 243 - 243r
sum(5) = a(1 - r^5)/(1-r)
242 = a(1 - r^5)/(1-r) ---- #2
divide #2 by #1
1-r^5 = 242/243
r^5 =1 - 242/243
r^5 = 1/243
r^5 = 1/3^5
r = 1/3
back in #1
a = 243-243(1/3)
= 162
a = 162, r = 1/3
check:
terms are:
162 54 18 6 2 2/3 ...
sum(5) = 242
sum(of all) = 162/(1-1/3)
= 162/(2/3) = 243
To determine the values of the common ratio and the first term of a convergent series, you can use the information given about the sum to infinity and the sum of the first few terms.
Step 1: Understand the scenario
- The sum to infinity of a convergent series is 243, meaning that if you add up all the terms in the series indefinitely, the total will be 243.
- The sum of the first five terms is given as 242.
Step 2: Write the formulas
- The formula for the sum to infinity of a convergent geometric series is:
sum = a / (1 - r)
Where "sum" is the sum to infinity, "a" is the first term, and "r" is the common ratio.
- The formula for the sum of the first "n" terms of a geometric series is:
sum = a * (1 - r^n) / (1 - r)
Step 3: Set up equations
- Use the given information to set up two equations:
Equation 1: 243 = a / (1 - r)
Equation 2: 242 = a * (1 - r^5) / (1 - r)
Step 4: Solve the equations
- To solve the equations simultaneously, you can use substitution or elimination.
- In this case, substitution would be more convenient.
- Rearrange Equation 1 to solve for "a":
a = 243 * (1 - r)
- Substitute the value of "a" in Equation 2:
242 = 243 * (1 - r) * (1 - r^5) / (1 - r)
Step 5: Simplify and solve for "r"
- Multiply both sides of the equation by (1 - r) to eliminate the denominator:
242 * (1 - r) = 243 * (1 - r) * (1 - r^5)
- Expand and simplify:
242 - 242r = 243 - 243r - 243r + 243r^6
- Combine like terms:
242 - 242r = 243 - 486r + 243r^6
- Rearrange and simplify:
243r^6 - 244r + 1 = 0
Step 6: Solve for "r"
- This equation is a polynomial equation, and solving it manually may be complex.
- You can use numerical methods or a graphing calculator to find the approximate values of "r".
Step 7: Solve for "a"
- Once you have the value of "r", you can substitute it into Equation 1 or Equation 2 to find the value of "a".
To determine the values of the common ratio and the first term in a convergent series, we can use the formula for the sum of an infinite geometric series:
S = a / (1 - r),
where:
- S is the sum to infinity,
- a is the first term, and
- r is the common ratio.
Given that the sum to infinity, S, is 243, we can substitute it in the formula:
243 = a / (1 - r).
Next, we are given that the sum of the first five terms is 242. We can use this to find another equation. The sum of the first five terms in a geometric series is given by:
S5 = a(1 - r^5) / (1 - r).
Substituting the given values:
242 = a(1 - r^5) / (1 - r).
Now we have a system of two equations with two variables. We can solve this system to find the values of a and r.
Rearranging the equations, we can write the system as:
243(1 - r) = a,
242(1 - r) = a(1 - r^5).
Now we can equate the expressions for a:
243(1 - r) = 242(1 - r^5).
Simplifying further, we have:
243 - 243r = 242 - 242r^5.
Now isolate the variable:
243r = 242r^5.
Divide both sides by r:
243 = 242r^4.
Divide both sides by 242:
243/242 = r^4.
Now take the fourth root of both sides:
(r^4)^(1/4) = (243/242)^(1/4).
Simplifying,
r = (243/242)^(1/4).
Now we can substitute the value of r back into one of the original equations to find the value of a. For simplicity, let's use the equation:
243 = a / (1 - r).
Substituting the value of r, we have:
243 = a / (1 - (243/242)^(1/4)).
Now we can solve for a by multiplying both sides by (1 - r):
243(1 - (243/242)^(1/4)) = a.
Simplify the equation, and you will find the value of a.
Therefore, by solving this system of equations, we can determine the values of the common ratio (r) and the first term (a).