Three numbers form a g.p.if the first and third numbers are 5and245 respectively,find two possible values for the middle number.
5, 35, 245
can you find the other one?
To find the middle number in a geometric progression (g.p.), we need to understand the relationship between the numbers in the progression. In a g.p., each term is obtained by multiplying the previous term by a fixed non-zero number called the common ratio (r).
Given that the first number (a₁) is 5 and the third number (a₃) is 245, we can construct the g.p. as follows:
a₁ = 5
a₃ = 245
We can express the terms of the g.p. using the formula:
aₙ = a₁ * r^(n-1)
Where aₙ represents the nth term of the g.p. and r represents the common ratio.
Let's find the common ratio (r) first, using the third term and the first term:
a₃ = a₁ * r^(3-1)
245 = 5 * r²
To isolate r, divide both sides of the equation by 5:
r² = 245/5
r² = 49
r = ±√49
So, the common ratio (r) can be either +7 or -7.
Now, to find the middle number, we need to find the second term (a₂). We can use the first term and the common ratio:
a₂ = a₁ * r^(2-1)
a₂ = a₁ * r
Substituting the values we know:
For r = +7:
a₂ = 5 * 7
a₂ = 35
For r = -7:
a₂ = 5 * (-7)
a₂ = -35
Hence, two possible values for the middle number in the geometric progression are 35 and -35.