1. If the 4th term and first term of an AP are 9 and 21 respectively. Find common difference.
We know that:
a4 = a1 + (4-1)d
Substituting the given values:
9 = 21 + 3d
Simplifying:
-12 = 3d
Therefore:
d = -4
The common difference is -4.
To find the common difference of an arithmetic progression (AP), we need the formula:
\[a_n = a_1 + (n-1)d\]
Where:
- \(a_n\) is the nth term of the AP
- \(a_1\) is the first term of the AP
- \(n\) is the position of the term in the AP
- \(d\) is the common difference of the AP
Given that the 4th term (\(a_4\)) of the AP is 9 and the first term (\(a_1\)) is 21, we can use the formula to solve for the common difference \(d\).
Let's substitute the values into the formula:
\[9 = 21 + (4-1)d\]
Simplifying the equation:
\[9 = 21 + 3d\]
Now, let's isolate \(d\) by subtracting 21 from both sides of the equation:
\[9 - 21 = 21 - 21 + 3d\]
\[-12 = 3d\]
Finally, let's solve for \(d\) by dividing both sides of the equation by 3:
\[-12/3 = 3d/3\]
\[-4 = d\]
Therefore, the common difference of the arithmetic progression is -4.