the 8th term of an a.p is 5times than third term while the 7th term is 9 greater than 4th term,write the first five terms of the a.p
the 8th term of an a.p is 5times than third term
a+7d = 5(a+2d)
7th term is 9 greater than 4th term
a+6d = 9+a+3d
Now just solve for a and d, and then you can write the AP.
To find the first five terms of an arithmetic progression (AP), we need to find the common difference (d) and the first term (a1).
Given information:
8th term (a8) = 5 times the 3rd term (a3)
7th term (a7) = 9 greater than the 4th term (a4)
Step 1: Finding the common difference (d)
We can find the common difference (d) using the formula:
d = (a8 - a7) = (a7 - a6) = (a6 - a5) = (a5 - a4) = (a4 - a3)
Step 2: Finding the 3rd term (a3)
We know that a8 = 5a3, so we can rewrite it as a8/5 = a3.
Step 3: Finding the 4th term (a4)
We know that a7 = a4 + 9, so we can rewrite it as a4 = a7 - 9.
Step 4: Finding the first term (a1)
We can find the first term (a1) using the formula:
a1 = a4 - 3d, where d is the common difference.
Step 5: Finding the first five terms
Using the values we found, we can calculate the first five terms of the AP.
a1 = a4 - 3d
a1 = (a7 - 9) - 3d
Now, we have all the information needed to calculate the first five terms of the AP.
The five terms can be written as: a1, a2, a3, a4, a5.
Please provide the common difference (d) or any one of the known terms (a3, a4, a7, a8) to proceed with the calculation.
To find the first five terms of an arithmetic progression (AP), we need two pieces of information: the common difference (d) and the value of any term in the sequence. In this case, we are given two conditions: the 8th term is 5 times the 3rd term, and the 7th term is 9 greater than the 4th term.
Let's solve these conditions step by step to find the common difference (d) and an arbitrary term in the sequence.
Given:
8th term = 5 * 3rd term
7th term = 4th term + 9
To find the common difference (d), we can compute the difference between the 8th and 3rd terms and the 7th and 4th terms.
8th term - 3rd term = 5 * 3rd term - 3rd term
7th term - 4th term = 4th term + 9 - 4th term
Simplifying these expressions:
5 * 3rd term - 3rd term = 8th term - 3rd term
4th term + 9 - 4th term = 7th term - 4th term
Combining like terms:
4 * 3rd term = 8th term - 3rd term
9 = 7th term - 4th term
From the first equation, we can conclude that the common difference (d) is equal to the difference between the 8th and 3rd terms:
d = 8th term - 3rd term
Using the given information, we substitute the values into this equation:
d = 5 * 3rd term - 3rd term
Simplifying:
d = 2 * 3rd term
Now, substituting the value of the common difference (d) into the second equation, we have:
9 = 7th term - 4th term
To eliminate the common difference, we substitute d = 2 * 3rd term:
9 = 7th term - (4th term + d)
9 = 7th term - 4th term - d
9 = 7th term - 4th term - 2 * 3rd term
Now we have two equations:
d = 2 * 3rd term
9 = 7th term - 4th term - 2 * 3rd term
We can use these equations to find the values of d and the terms in the AP.
Solving the first equation for d:
d = 2 * 3rd term
d = 6th term
Substituting this result into the second equation:
9 = 7th term - 4th term - 6th term
9 = 7th term - 10th term
Now, let's find the values of the terms by using any variable. Let's assume the first term (a) to be "a" and the common difference (d) to be 6:
1st term (a) = a
2nd term = a + d = a + 6
3rd term = a + 2d = a + 2 * 6 = a + 12
4th term = a + 3d = a + 3 * 6 = a + 18
5th term = a + 4d = a + 4 * 6 = a + 24
Therefore, the first five terms of this AP are:
1st term = a
2nd term = a + 6
3rd term = a + 12
4th term = a + 18
5th term = a + 24