The 3rd term of an ap is 9 while the 11th term is -7,find the first five term of ap
To find the first five terms of an arithmetic progression (AP), we need two pieces of information: the common difference (d) and either the first term (a₁) or any other term in the sequence.
Given that the 3rd term is 9 and the 11th term is -7, we can use this information to find both the common difference and the first term.
Step 1: Finding the common difference (d)
We can use the formula for the nth term of an AP to find the common difference:
aₙ = a₁ + (n-1)d
Using the information given, we can plug in the values:
9 = a₁ + (3-1)d
-7 = a₁ + (11-1)d
Simplifying these equations, we get two equations:
a₁ + 2d = 9 ...(1)
a₁ + 10d = -7 ...(2)
Step 2: Solving the equations to find the common difference (d)
To find the common difference (d), we can subtract equation (1) from equation (2):
a₁ + 10d - (a₁ + 2d) = -7 - 9
8d = -16
d = -16/8
d = -2
Step 3: Finding the first term (a₁)
We can substitute the value of d into equation (1):
a₁ + 2(-2) = 9
a₁ - 4 = 9
a₁ = 9 + 4
a₁ = 13
Now that we have the common difference (d = -2) and the first term (a₁ = 13), we can find the first five terms of the AP.
Term 1: a₁ = 13
Term 2: a₂ = a₁ + d = 13 + (-2) = 11
Term 3: a₃ = a₁ + 2d = 13 + 2(-2) = 13 - 4 = 9
Term 4: a₄ = a₁ + 3d = 13 + 3(-2) = 13 - 6 = 7
Term 5: a₅ = a₁ + 4d = 13 + 4(-2) = 13 - 8 = 5
Therefore, the first five terms of the arithmetic progression are 13, 11, 9, 7, and 5.
T11-T3 = 8d = (-7)-9 = -16
so, d = -2
Now you can find a, and then write the 1st 5 terms.
Or, knowing d=-2, and T3=9, you can work from both sides of T3:
13 11 9 7 5