calculate the first ten term of an Arithmetic progression whose 5th term is 24 and the difference the 7th and 10th term is 15
A10-A7 = 3d = 15
Now you know d, so you can get a, and then the terms.
4, 9, 14, 19, 24, 29, 34, 39, 44, 49
To find the first ten terms of an arithmetic progression, we first need to find the common difference (d).
We are given two pieces of information:
1. The 5th term is 24.
2. The difference between the 7th and 10th terms is 15.
Let's use this information to find the common difference:
Step 1: Finding the common difference (d)
Let the 5th term be a + 4d (since it's the 5th term, we add 4d)
Let the 7th term be a + 6d (since it's the 7th term, we add 6d)
Let the 10th term be a + 9d (since it's the 10th term, we add 9d)
Given that the difference between the 7th and 10th terms is 15:
(a + 9d) - (a + 6d) = 15
a + 9d - a - 6d = 15
3d = 15
d = 15/3
d = 5
So, the common difference, d, is 5.
Step 2: Finding the first ten terms:
The formula to find the nth term of an arithmetic progression is:
nth term (an) = a + (n-1)d
where a is the first term and d is the common difference.
Using the given information that the 5th term is 24, we can now find the first term, a:
a + (5-1)5 = 24
a + 4(5) = 24
a + 20 = 24
a = 24 - 20
a = 4
Now we can find the first ten terms:
1st term (a) = 4
2nd term (a + d) = 4 + 5 = 9
3rd term (a + 2d) = 4 + 2(5) = 14
4th term (a + 3d) = 4 + 3(5) = 19
5th term (a + 4d) = 4 + 4(5) = 24
6th term (a + 5d) = 4 + 5(5) = 29
7th term (a + 6d) = 4 + 6(5) = 34
8th term (a + 7d) = 4 + 7(5) = 39
9th term (a + 8d) = 4 + 8(5) = 44
10th term (a + 9d) = 4 + 9(5) = 49
Therefore, the first ten terms of the arithmetic progression are:
4, 9, 14, 19, 24, 29, 34, 39, 44, 49.
To find the first ten terms of an arithmetic progression, we first need to determine the common difference of the progression.
Let's say the first term is "a" and the common difference is "d". We can use the formula for the nth term of an arithmetic progression to find the value of "a":
nth term = a + (n - 1) * d
Given that the 5th term is 24, we can substitute the values into the formula:
24 = a + (5 - 1) * d
24 = a + 4d
Similarly, we can use the formula for the 7th term and the 10th term to create two more equations:
7th term = a + (7 - 1) * d
10th term = a + (10 - 1) * d
Since the difference between the 7th and 10th terms is 15, we can substitute the values into the equation:
10th term - 7th term = 15
(a + (10 - 1) * d) - (a + (7 - 1) * d) = 15
9d - 6d = 15
3d = 15
d = 5
Now that we know the value of the common difference is 5, we can substitute it back into the equation we found earlier:
24 = a + 4 * 5
24 = a + 20
a = 24 - 20
a = 4
Therefore, the first term (a) is 4 and the common difference (d) is 5.
To calculate the first ten terms, we can use the formula for the nth term of an arithmetic progression:
nth term = a + (n - 1) * d
Substituting the values we found:
1st term = 4 + (1 - 1) * 5 = 4
2nd term = 4 + (2 - 1) * 5 = 9
3rd term = 4 + (3 - 1) * 5 = 14
4th term = 4 + (4 - 1) * 5 = 19
5th term = 4 + (5 - 1) * 5 = 24
6th term = 4 + (6 - 1) * 5 = 29
7th term = 4 + (7 - 1) * 5 = 34
8th term = 4 + (8 - 1) * 5 = 39
9th term = 4 + (9 - 1) * 5 = 44
10th term = 4 + (10 - 1) * 5 = 49
Therefore, the first ten terms of the arithmetic progression are:
4, 9, 14, 19, 24, 29, 34, 39, 44, 49.