The 5th term of the expontential sequence is 4375 and 2nd term is 35,
(a) Find the 6th term
(b) Sum of the five term
Good
since T5 = 3 terms after T3, you have
r^3 = 4375/35 = 125 = 5^3
So, use that to find a, and then recall that
S5 = a (r^5-1)/(r-1)
(a) Okay, let me put on my "math clown" hat for this one! To find the 6th term of the exponential sequence, we can use the formula:
nth term = a * r^(n-1),
where a is the first term and r is the common ratio.
Since we know that the 2nd term is 35, we can substitute these values into the formula:
35 = a * r^(2-1).
Simplifying, we get:
35 = a * r.
Now, we also have the 5th term, which is 4375. Substituting these values into the formula again:
4375 = a * r^(5-1).
Simplifying further:
4375 = a * r^4.
Now we have two equations:
35 = a * r,
4375 = a * r^4.
Dividing the second equation by the first equation, we get:
4375/35 = r^4.
Solving for r, we find:
r^4 = 125.
Taking the fourth root of both sides, we get:
r = 5.
Now, substituting this value back into the first equation:
35 = a * 5.
Solving for a, we find:
a = 7.
So, the 6th term can be found by substituting a = 7, r = 5, and n = 6 into the formula:
6th term = a * r^(n-1) = 7 * 5^(6-1) = 7 * 5^5.
(b) To find the sum of the five terms, we can use the formula for the sum of a geometric series:
Sum = a * (r^n - 1) / (r - 1).
Substituting the given values:
Sum = 7 * (5^5 - 1) / (5 - 1).
Now it's just a matter of calculating this expression to get the sum!
To find the 6th term of the exponential sequence, we need to determine the common ratio. Let's call the first term of the sequence "a" and the common ratio "r".
Given that the 2nd term is 35 and the 5th term is 4375, we can write two equations based on the formula for an exponential sequence:
(a)(r)^(2-1) = 35 (1)
(a)(r)^(5-1) = 4375 (2)
From equation (1), we have:
(a)(r) = 35
Divide equation (2) by equation (1) to eliminate "a":
[(a)(r)^(5-1)] / [(a)(r)^(2-1)] = 4375 / 35
Simplifying the equation further, we have:
(r)^4 = 125
Taking the fourth root of both sides, we get:
r = ∛(125)
Simplifying ∛(125), we find:
r = 5
Now that we know the common ratio (r = 5), we can find the first term (a) by substituting the values into equation (1):
(a)(5) = 35
Dividing both sides by 5, we get:
a = 7
(a) The 6th term can be calculated using the formula for an exponential sequence:
6th term = (a)(r)^(6-1)
Substituting the values we found:
6th term = (7)(5)^(5)
Calculating further:
6th term = 7 * 5^5
6th term = 7 * 3125
6th term = 21875
Therefore, the 6th term of the exponential sequence is 21875.
(b) To find the sum of the five terms, we can use the formula for the sum of a finite exponential sequence:
Sum = a * (r^n - 1) / (r - 1)
Substituting the known values:
Sum = 7 * (5^5 - 1) / (5 - 1)
Calculating further:
Sum = 7 * (3125 - 1) / 4
Sum = 7 * 3124 / 4
Sum = 7 * 781
Sum = 5467
Therefore, the sum of the five terms is 5467.
To find the 6th term of the exponential sequence, we need to determine the common ratio.
The exponential sequence can be represented as: a * r^(n-1), where:
- a is the first term,
- r is the common ratio, and
- n is the term number.
We are given that the 2nd term is 35, so we can write the equation as: a * r^(2-1) = 35. This gives us the equation: a * r = 35.
We are also given that the 5th term is 4375. Using the formula, we can write the equation as: a * r^(5-1) = 4375. Simplifying, we get: a * r^4 = 4375.
Now we have two equations: a * r = 35 and a * r^4 = 4375. We can solve these equations simultaneously to find the values of a and r.
Dividing the second equation by the first equation, we get: (a * r^4) / (a * r) = 4375/35. Simplifying, we have: r^3 = 125. Since 125 is the cube of 5, we can conclude that r = 5.
Now that we have the value of r, we can substitute it into one of the earlier equations to find the value of a. Using the equation a * r = 35, we substitute r = 5: a * 5 = 35. Solving for a, we find that a = 7.
(a) To find the 6th term, we can use the equation a * r^(n-1). Plugging in the values a = 7, r = 5, and n = 6, we have: 7 * 5^(6-1). Evaluating this expression, we find that the 6th term is 875.
(b) To find the sum of the five terms, we can use the formula for the sum of a finite geometric series: S = a * (1 - r^n) / (1 - r), where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.
Plugging in the values a = 7, r = 5, and n = 5, we have: S = 7 * (1 - 5^5) / (1 - 5). Evaluating this expression, we find that the sum of the five terms is 874.