The sixth term of an AP is twice the 3rd term and the first term is 3 . Find the common difference and the 10th term
a+5d = 2(a+2d)
a+2d = 3
Now solve for a and d, then find a+9d
oops. That's a=3, not a+2d
Good
To solve this problem, we'll use the formula for the nth term of an arithmetic progression (AP):
nth term = first term + (n - 1) * common difference
Let's denote the common difference as 'd'. Given that the first term (a1) is 3, we can express the third term (a3) and sixth term (a6) in terms of 'd':
a3 = a1 + (3 - 1) * d = 3 + 2d
a6 = a1 + (6 - 1) * d = 3 + 5d
We are given that the sixth term (a6) is twice the third term (a3), so we can write the equation:
a6 = 2 * a3
Substituting the expressions for a3 and a6, we have:
3 + 5d = 2 * (3 + 2d)
Simplifying the equation:
3 + 5d = 6 + 4d
Rearranging the terms:
5d - 4d = 6 - 3
d = 3
The common difference is 3.
To find the 10th term, we substitute n = 10, a1 = 3, and d = 3 into the formula for the nth term:
a10 = 3 + (10 - 1) * 3
= 3 + 9 * 3
= 3 + 27
= 30
Therefore, the 10th term is 30.