The 6th term of an AP is twice the third term.what is the common difference?
a+5d = 2(a+2d)
a+5d = 2a+4d
a = d
So, pick a first term, and use that for the difference. The sequence is thus
a,2a,3a,4a,5a,6a,...
6a = 2*3a
To find the common difference of an arithmetic progression (AP) where the 6th term is twice the third term, you can follow these steps:
Step 1: Understand the problem.
In an arithmetic progression, each term is obtained by adding a fixed value, called the common difference, to the previous term. The 6th term is the value of the AP that occurs in the 6th position, while the third term is the value that occurs in the third position. We need to find the common difference.
Step 2: Identify and label the necessary values.
Let's label the third term as "a" and the common difference as "d". The 6th term is twice the third term, so it can be represented as 2a.
Step 3: Write the formula for the nth term of an AP.
The formula for the nth term of an AP is given by: an = a1 + (n - 1)d, where an represents the nth term, a1 denotes the first term, and d is the common difference.
Step 4: Apply the given information to the formula.
Using the formula, we have:
a6 = a1 + (6 - 1)d, and
a3 = a1 + (3 - 1)d.
We know that a6 is twice a3, so we can write the equation as:
2a3 = a1 + (6 - 1)d.
Step 5: Solve the equations simultaneously.
By substituting the equations for a6 and a3 into the equation representing the condition that the 6th term is twice the third term, we get:
2a3 = a1 + 5d.
Now, we can equate the two equations involving a3:
a1 + (3 - 1)d = a1 + 5d.
Step 6: Simplify and solve for the common difference.
Simplifying the equation:
2d = 4d.
Subtracting 4d from both sides:
2d - 4d = 4d - 4d,
-2d = 0.
Dividing by -2, we get:
d = 0.
Therefore, the common difference of the arithmetic progression is 0.