the 2nd term of a g.p is 12 more than the 1st term,given that the common ratio is half of the 1st term find the third term of the g.p
Given:
t2 = t1 + 12.......Eq 1
r = (1/2) × t1........Eq 2
Solution:
Since it's a geometric progression, we know that:
t2 = t1 × r ........Eq 3
Substitute Eq 2 into Eq 3:
t2 = t1 × (1/2) × t1 ....... Eq 4
Substitute Eq 1 into Eq 4:
t1 + 12 = t1 × (1/2) × t1
Solve for t1, and you will now have enough info to find t3.
Pls how do I solve for t1
To find the third term of the geometric progression (G.P.), the information given is:
1. The second term is 12 more than the first term.
2. The common ratio is half of the first term.
Let's break down the steps to find the third term:
Step 1: Define the terms of the G.P.
Let the first term be denoted as 'a', the second term as 'ar', and the third term as 'ar^2'. Here, 'a' represents the first term, 'r' represents the common ratio, and 'ar^2' represents the third term.
Step 2: Express the given information in terms of 'a' and 'r'.
According to the given information, the second term is 12 more than the first term:
ar = a + 12
Also, the common ratio is half of the first term:
r = a/2
Step 3: Solve the equations simultaneously.
Now, substitute the value of 'r' from the second equation into the first equation:
(a/2)a = a + 12
Simplifying the equation further:
a^2/2 = a + 12
Multiply both sides by 2 to get rid of the fraction:
a^2 = 2a + 24
Rearrange the equation in a quadratic form:
a^2 - 2a - 24 = 0
Factorize the expression:
(a - 6)(a + 4) = 0
Applying the zero product property:
a - 6 = 0 or a + 4 = 0
So, two possible values for 'a' are:
a = 6 or a = -4
Step 4: Find the value of 'r' using the value of 'a'.
Now, substitute the value of 'a' back into the second equation to find 'r':
For a = 6:
r = a/2 = 6/2 = 3
For a = -4:
r = a/2 = -4/2 = -2
Step 5: Calculate the third term (ar^2) for both values of 'a'.
For a = 6:
Third term (ar^2) = (6)(3^2) = 54
For a = -4:
Third term (ar^2) = (-4)(-2^2) = -16
Hence, the third term of the geometric progression can be either 54 or -16, depending on the possible values of the first term 'a'.