first term and larst term of a geometric progression is 42.if the forth term is greater than the second term by 168, find the first term.the forth term.
check your typing.
If the first and last term of a GS are the same, then r = 1, that is, the terms do not change.
But then how can there be a difference between the fourth and second term?
Did you mean..
"The sum of the 1st and last terms is 42" ?
Either way, you would have a major problem:
you have 3 unknowns, a, r, and n
but only two sets of information.
first term and larst term of a geometric progression is 42.if the forth term is greater than the second term by 168, find the first term.the forth term
Please help me with the answer
To find the first term and the fourth term of a geometric progression, we need to use the properties of a geometric sequence.
Let's suppose that the first term is 'a' and the common ratio is 'r'.
We are given two conditions:
1. The first term and the last term of the sequence is 42. This means that the last term of the sequence is also 42. Since the last term of a geometric progression is given by the formula:
last term = first term * (common ratio)^(number of terms - 1)
We can write this as:
42 = a * r^(n-1) ----(1)
2. The fourth term is greater than the second term by 168. This means that:
a * r^(4-1) = a * r^3 (fourth term)
a * r^(2-1) = a * r^1 (second term)
The difference between the fourth and second term can be expressed as:
a * r^3 - a * r^1 = 168 ----(2)
From the given information, we have two equations (Equation 1 and Equation 2) and two unknowns (a and r). We can solve these equations to find the values of 'a' and 'r'.
Step 1: Divide Equation 2 by Equation 1 to eliminate 'a':
(a * r^3 - a * r^1) / (a * r^(n-1)) = 168 / 42
Simplifying this expression:
r^3 - r = 4
Step 2: Factorize the expression:
r(r^2 - 1) = 4
Step 3: Apply the difference of squares formula:
r(r + 1)(r - 1) = 4
Step 4: Find the possible values of 'r' that satisfy the equation. In this case, we can see that r = 2 is a possible solution.
Now, let's substitute the value of 'r' = 2 into one of the previous equations. Let's use Equation 1:
42 = a * 2^(n-1)
Since n is not mentioned, we cannot solve for 'a' and 'n' specifically. However, we can find the first term and the fourth term.
If n = 4 (in order to find the fourth term), we have:
42 = a * 2^(4-1)
42 = a * 2^3
42 = 8a
Solving for 'a' gives:
a = 42 / 8
a = 5.25
Therefore, the first term is 5.25 and the fourth term can be found by substituting 'a' and 'r' into the formula for the fourth term:
fourth term = a * r^(4-1)
= 5.25 * 2^3
= 5.25 * 8
= 42
So, the fourth term is 42.