The fifth term of an G.P is 4375 and second term is 35.Find the first term
d = (4375-35)/(5-2)
a = 35-d
my bad. I read AP
r = 5
so a = 7
To find the first term of a geometric progression (G.P.), we can use the formula:
\[a_n = a_1 \cdot r^{(n-1)}\]
Where:
- \(a_n\) is the \(n\)th term of the G.P.
- \(a_1\) is the first term of the G.P.
- \(r\) is the common ratio between consecutive terms.
- \(n\) is the position of the term in the G.P.
Given that the second term of the G.P. is 35, we can substitute the values:
\[35 = a_1 \cdot r^{(2-1)}\]
Simplifying, we have:
\[35 = a_1 \cdot r\]
Now, given that the fifth term of the G.P. is 4375, we can substitute the values:
\[4375 = a_1 \cdot r^{(5-1)}\]
Simplifying, we have:
\[4375 = a_1 \cdot r^4\]
Now, we have two equations:
1) \(35 = a_1 \cdot r\)
2) \(4375 = a_1 \cdot r^4\)
To find the first term (\(a_1\)), we can divide equation 2) by equation 1):
\(\frac{4375}{35} = \frac{a_1 \cdot r^4}{a_1 \cdot r}\)
Simplifying, we have:
\(125 = r^3\)
Taking the cube root of both sides:
\(r = \sqrt[3]{125}\)
\(r = 5\)
Substitute this value of \(r\) back into equation 1):
\(35 = a_1 \cdot 5\)
To find \(a_1\), divide both sides by 5:
\(\frac{35}{5} = a_1\)
Simplifying, we have:
\(7 = a_1\)
Therefore, the first term of the G.P. is 7.
To find the first term of a geometric progression (G.P.), we can use the formula:
an = a1 * r^(n - 1)
Where:
an = the nth term of the G.P.
a1 = the first term of the G.P.
r = the common ratio of the G.P.
n = the position of the term in the G.P.
In this case, we are given:
a5 = 4375 (the fifth term)
a2 = 35 (the second term)
Let's calculate the common ratio (r):
To find the common ratio, we can use the formula:
r = a2 / a1
So, substituting the given values:
r = 35 / a1
Now, we can find the value of a1 (the first term) using the equation for the fifth term:
a5 = a1 * r^(5 - 1)
Substituting the value of a5 = 4375 and r = 35 / a1:
4375 = a1 * (35 / a1)^(4)
To solve this equation, we need to simplify the right side expression:
4375 = 35^4 / a1^3
Now, to solve for a1, we can cross multiply:
a1^3 * 4375 = 35^4
Rearranging the equation:
a1^3 = 35^4 / 4375
Simplifying further:
a1^3 = (35^2)^2 / (5^3)
a1^3 = 1225^2 / 125
Taking the cube root of both sides:
a1 = (1225^2 / 125)^(1/3)
Using a calculator, we can evaluate this expression to find the first term, a1. The approximate value of a1 is 7.