Two APS have the same first and last terms.The first AP has 21 terms with a common differences of 9. How many terms has the other ap if its common differences is 4.
first AS:
first term: a
d = 9
n = 21
term(21) = a + 9(20) = a + 180
2nd AS
first term : a
d = 4
term(n) = a + 4(n-1)
a + 4(n-1) = a + 180
4n - 4 = 180
4n = 184
n = 46 <----- number of terms for the 2nd sequence
check
term(21) for the first = a + 180
term(46) for the second = a + 4(45) = a + 180 , the same as the last term of sequence #1
looks like the first term could be anything, as long as it is the same for both
To find the number of terms in the second arithmetic progression (AP) with a common difference of 4, we can use the formula for the nth term of an AP:
nth term = first term + (n-1) * common difference
We are given that the first and last terms of both APs are the same. Let's denote this common value as "x".
For the first AP with 21 terms and a common difference of 9, we can write the equation for the last term as:
x = first term + (21-1) * 9
Simplifying this equation:
x = first term + 180
For the second AP with an unknown number of terms and a common difference of 4, we can write the equation for the last term as:
x = first term + (n-1) * 4
Since the first and last terms of both APs are the same, we can equate the right sides of both equations:
first term + 180 = first term + (n-1) * 4
Simplifying this equation:
180 = (n-1) * 4
Dividing both sides by 4:
45 = n-1
Adding 1 to both sides:
n = 46
Therefore, the second AP has 46 terms with a common difference of 4.
To solve this question, we need to understand the concept of an arithmetic progression (AP).
An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference.
Let's determine the first term and last term of the first AP mentioned in the question.
The first term of the AP is denoted by 'a', and the common difference is denoted by 'd'.
Given that the first AP has 21 terms with a common difference of 9, we can find the first term as follows:
a = first term = ?
d = common difference = 9
n = number of terms = 21
The formula to find the nth term of an AP is given by: an = a + (n-1)d
Substituting the given values, we can find the first term of the AP:
a = a + (21-1) * 9
a = a + 180
Since the first and last terms of the first AP are the same, let's denote them as 'x'. Therefore, we have:
x = x + 180
Simplifying this equation, we find:
180 = 0
This equation has no solution because the left-hand side is a constant value, and the right-hand side is zero, which means there is a contradiction.
Hence, the given information in the question is incorrect.
Therefore, we cannot determine the number of terms in the other AP based on the information provided.