The 12th term of an AP is four times 4th term what is the relationship between the first term
Well, let's see. The relationship between the first term and the 12th term in an arithmetic progression (AP) depends on the common difference of the sequence. If we take the common difference as "d", then we can express the 12th term as follows:
12th term = 1st term + (12 - 1)*d (since the n-th term of an AP is given by Tn = a + (n - 1)*d)
And if the 12th term is four times the 4th term, we can write it as:
12th term = 4th term * 4
Now, if we substitute the expressions for both the 12th term and the 4th term, we get:
1st term + (12 - 1)*d = (4th term * 4)
Simplifying this equation will give us the relationship between the first term and the common difference, which in turn determines the entire AP.
a+11d = 4(a+3d)
To find the relationship between the first term and the common difference of an arithmetic progression (AP) based on the 12th term being four times the 4th term, we can use the formula for the nth term of an AP.
The nth term of an AP can be calculated using the formula:
Tn = a + (n-1)d
where Tn is the nth term, a is the first term, n is the position of the term, and d is the common difference.
Given that the 12th term (T12) is four times the 4th term (T4), we can write the equation:
T12 = 4 * T4
Using the formula for the nth term, we have:
a + (12-1)d = 4 * (a + (4-1)d)
Simplifying the equation:
a + 11d = 4a + 9d
Combine like terms:
-3a = -2d
Divide both sides by -2:
a/d = 3/2
Therefore, the relationship between the first term and the common difference in this arithmetic progression is that the first term is 3/2 times the common difference.
To find the relationship between the first term of an arithmetic progression (AP) and the 12th term, you can make use of the formula for the nth term of an AP.
The formula for the nth term of an AP is given by:
Tn = a + (n-1)d
Where:
Tn = nth term
a = first term
n = the term number
d = common difference between terms
Given that the 12th term (T12) is four times the 4th term (T4), we can write the following equation:
T12 = 4 * T4
Now, we'll substitute the nth term formula into this equation:
a + (12-1)d = 4 * (a + (4-1)d)
Simplifying this equation, we get:
a + 11d = 4a + 3d
Rearranging the terms and combining like terms, we have:
11d - 3d = 4a - a
8d = 3a
Therefore, the relationship between the first term (a) and the common difference (d) is 8d = 3a.