In a gp fifth is 4 times the 3rd term and sum of first two terms is -4. Find the terms of gp
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If the 5th is 4 times the 3rd, then r=2
a+ar = -4
a+2a = -4
a = -4/3
now take it from there.
To find the terms of the GP (Geometric Progression), let's begin by assigning variables.
Let the first term be "a" and the common ratio be "r".
We're given two conditions:
1) The fifth term is 4 times the third term:
The third term can be calculated as a * r².
The fifth term can be calculated as a * r⁴.
According to the given information, the fifth term (a * r⁴) is equal to 4 times the third term (4 * a * r²):
a * r⁴ = 4 * a * r².
Dividing both sides of the equation by "a" (assuming "a" is not equal to 0) gives us:
r⁴ = 4 * r².
Dividing both sides by "r²" (assuming "r" is not equal to 0) gives us:
r² = 4.
Taking the square root of both sides (ignoring the negative root since we are looking for a positive common ratio) gives us:
r = 2.
2) The sum of the first two terms is -4:
The sum of the first two terms can be calculated as a + a * r (a + ar).
According to the given information, the sum of the first two terms (a + a * r) is equal to -4:
a + a * r = -4.
Substituting the value of "r" (2) into the equation gives us:
a + a * 2 = -4.
Simplifying further:
a + 2a = -4,
3a = -4,
a = -4/3.
Now we have found the first term (a = -4/3) and the common ratio (r = 2) of the GP.
Therefore, the terms of the GP are:
First term (a) = -4/3,
Second term = a * r = (-4/3) * 2 = -8/3,
Third term = a * r² = (-4/3) * (2²) = -4,
Fourth term = a * r³ = (-4/3) * (2³) = -32/3,
Fifth term = a * r⁴ = (-4/3) * (2⁴) = -64/3.
So, the terms of the GP are: -4/3, -8/3, -4, -32/3, and -64/3.