The 3rd term of an ap is 9 while the 11th term is -7 find the first five term of the ap
a+2d = 9
a+11d = -7
Now solve for a and d.
Or, knowing that there are 8 terms from the 3rd to the 11th,
8d = -16
and you get d easily
now get a and let 'er rip
To find the first five terms of an arithmetic progression (AP), we need to determine the common difference between the terms, which remains constant throughout the progression.
Given that the 3rd term is 9 and the 11th term is -7, we can use the formula for the nth term of an AP to find the common difference (d). The formula is:
an = a1 + (n-1)d
Where:
an = nth term of the AP
a1 = first term of the AP
d = common difference between the terms
n = position of the term in the AP
Using the information provided:
For the 3rd term:
9 = a1 + (3-1)d ...(1)
For the 11th term:
-7 = a1 + (11-1)d ...(2)
We can solve these two equations simultaneously to find the common difference (d) and the first term (a1).
Subtracting equation (1) from equation (2), we get:
-7 - 9 = a1 + (11-1)d - (a1 + (3-1)d)
-16 = 8d
Dividing both sides by 8, we find that:
d = -2
Substituting the common difference (d) into equation (1), we have:
9 = a1 + (3-1)(-2)
9 = a1 - 4
Adding 4 to both sides, we find that:
a1 = 13
Therefore, the first term (a1) of the arithmetic progression is 13, and the common difference (d) is -2.
Now we can find the first five terms of the AP using the formula:
an = a1 + (n-1)d
Substituting the values, we have:
1st term (a1) = 13
2nd term = a1 + (2-1)d = 13 + (2-1)(-2) = 13 - 2 = 11
3rd term = a1 + (3-1)d = 13 + (3-1)(-2) = 13 - 4 = 9
4th term = a1 + (4-1)d = 13 + (4-1)(-2) = 13 - 6 = 7
5th term = a1 + (5-1)d = 13 + (5-1)(-2) = 13 - 8 = 5
Therefore, the first five terms of the AP are: 13, 11, 9, 7, 5.
To find the first five terms of an Arithmetic Progression (AP), we need to determine the common difference (d) first. Once we know the common difference, we can proceed to find the first five terms.
The formula for the nth term of an AP is given by:
an = a1 + (n - 1)d
Given information:
a3 = 9 (third term)
a11 = -7 (eleventh term)
Using the formula, we can set up the following equations:
a3 = a1 + 2d ... equation 1
a11 = a1 + 10d ... equation 2
From equation 1, we can rewrite a1 in terms of d:
a1 = a3 - 2d
Substituting this into equation 2, we get:
a11 = (a3 - 2d) + 10d
-7 = a3 + 8d
Rearranging the equation:
8d = -7 - a3
Substituting the given value of a3 = 9:
8d = -7 - 9
8d = -16
d = -2
Now that we have found the common difference (d = -2), we can find the first five terms using the formula an = a1 + (n - 1)d:
a1 = a3 - 2d = 9 - (-2)(2) = 9 + 4 = 13
Therefore, the first five terms of the AP are:
a1 = 13
a2 = a1 + d = 13 + (-2) = 11
a3 = a1 + 2d = 13 + (-2)(2) = 13 + (-4) = 9
a4 = a1 + 3d = 13 + (-2)(3) = 13 + (-6) = 7
a5 = a1 + 4d = 13 + (-2)(4) = 13 + (-8) = 5
Hence, the first five terms of the AP are: 13, 11, 9, 7, 5.