The fifth term of an geometric sequence is 4375 and the second term is 35.find (a)the third term (b)the sixth term (c)the first five term
35 r^3 = 4375
looks like the ratio (r) is 5
start with the 2nd term and keep multiplying by 5
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.
To find the solutions, we need to use the formula for the nth term of a geometric sequence:
An = A1 * r^(n-1)
where An is the nth term, A1 is the first term, r is the common ratio, and n is the position of the term in the sequence.
Given Information:
A2 = 35 (second term)
A5 = 4375 (fifth term)
First, let's find the common ratio (r):
A2 = A1 * r^(2-1)
35 = A1 * r
Now, let's find the first term (A1) using the obtained value of the common ratio:
A5 = A1 * r^(5-1)
4375 = A1 * r^4
We have two equations:
Equation 1: 35 = A1 * r
Equation 2: 4375 = A1 * r^4
(a) To find the third term:
Using Equation 1, divide Equation 2 by the square of Equation 1:
4375 = (35 * r^3) * r
4375 = 35r^4
r^4 = 4375/35
r^4 = 125
Now, take the fourth root of both sides to solve for r:
r = ∛(125)
r = 5
Now, substitute the value of r into Equation 1 to find A1:
35 = A1 * 5
A1 = 35/5
A1 = 7
Use the formula to find the third term (A3):
A3 = A1 * r^(3-1)
A3 = 7 * 5^2
A3 = 7 * 25
A3 = 175
Therefore, the third term (a) is 175.
(b) To find the sixth term:
Now, substitute the values of A1 and r into the formula to find A6:
A6 = A1 * r^(6-1)
A6 = 7 * 5^5
A6 = 7 * 3125
A6 = 21875
Therefore, the sixth term (b) is 21,875.
(c) To find the first five terms:
Using the formula, we can find the first five terms by substituting the values of A1 and r:
First term (A1): 7
Second term (A2): 7 * 5
Third term (A3): 7 * 5^2
Fourth term (A4): 7 * 5^3
Fifth term (A5): 7 * 5^4
Calculating these values will give us the first five terms:
A1 = 7
A2 = 7 * 5 = 35
A3 = 7 * 5^2 = 7 * 25 = 175
A4 = 7 * 5^3 = 7 * 125 = 875
A5 = 7 * 5^4 = 7 * 625 = 4375
Therefore, the first five terms (c) are 7, 35, 175, 875, and 4375.