The fifth term of an exponential sequence is 4375 and the second term is 35. Find a) the third term. b) the sixth term. c)sum of the first five terms.
ar^4 = 4375
ar = 35
divide ...
ar^4 / ar = 4375/35
r^3 = 125
r = 5
continue to find a ,
then you have formulas for all the required results
a) The third term of the exponential sequence can be found by applying the formula:
nᵗʰ term = a × r^(n-1)
Given that the second term is 35, we can substitute these into the formula:
35 = a × r^(2-1)
35 = a × r
Now, let's use the fifth term to form another equation:
4375 = a × r^(5-1)
4375 = a × r^4
Now, we can divide the two equations to eliminate the term 'a':
(4375/35) = (a × r^4) / (a × r)
125 = r^3
Taking the cube root of both sides, we find:
r = 5
Now that we know the value of 'r', we can substitute it back into one of the previous equations to find 'a':
35 = a × 5
a = 7
Therefore, the third term is given by:
3ᵗʰ term = a × r^(3-1)
= 7 × 5^2
= 7 × 25
= 175
b) The sixth term can be calculated using the same formula:
6ᵗʰ term = a × r^(6-1)
= 7 × 5^5
= 7 × 3125
= 21875
c) The sum of the first five terms can be found by adding them together:
Sum = 1ᵗʰ term + 2ᵗʰ term + 3ᵗʰ term + 4ᵗʰ term + 5ᵗʰ term
= a + a × r + a × r^2 + a × r^3 + a × r^4
= a × (1 + r + r^2 + r^3 + r^4)
= 7 × (1 + 5 + 25 + 125 + 625)
= 7 × 781
= 5467
Therefore, the sum of the first five terms is 5467.
answer
I don understand the question
Is that a or b
I want a and b now please
To find the missing terms and the sum of the first five terms, we need to determine the common ratio (r) of the exponential sequence.
Let's use the given data:
Second term (a2) = 35
Fifth term (a5) = 4375
We know that the nth term (an) formula in an exponential sequence is given by:
an = a1 * r^(n-1),
where a1 is the first term and n is the position of the term.
a) Finding the third term:
We have the second term (a2) = 35, and we need to find the third term (a3). Plugging in the values into the formula:
a2 = a1 * r^(2-1)
35 = a1 * r,
We don't have the explicit values of a1 or r, but we can proceed to find them by using the fifth term (a5).
b) Finding the sixth term:
We have the fifth term (a5) = 4375, and we need to find the sixth term (a6). Using the same approach:
a5 = a1 * r^(5-1)
4375 = a1 * r^4.
c) Finding the sum of the first five terms:
Using the formula for the sum of an exponential series:
Sn = a1 * (1 - r^n) / (1 - r) ,
where Sn is the sum of the first n terms.
We can calculate this by finding a1 and r using the given information and substituting them into the formula.
Let's calculate a1 and r using the information we have:
35 = a1 * r,
4375 = a1 * r^4.
Dividing these two equations:
(a1 * r^4) / (a1 * r) = 4375 / 35,
r^3 = 125,
r = ∛125 = 5.
Now, we can find a1 by substituting r = 5 into the initial equation:
35 = a1 * 5,
a1 = 35 / 5 = 7.
a) To find the third term:
a3 = a1 * r^(3-1),
a3 = 7 * 5^2 = 7 * 25 = 175.
b) To find the sixth term:
a6 = a1 * r^(6-1),
a6 = 7 * 5^5 = 7 * 3125 = 21875.
c) To find the sum of the first five terms:
S5 = a1 * (1 - r^5) / (1 - r),
S5 = 7 * (1 - 5^5) / (1 - 5) = 7 * (1 - 3125) / -4 = -7 * (-3124) / 4 = 54631.
Therefore:
a) The third term is 175.
b) The sixth term is 21875.
c) The sum of the first five terms is 54631.