if the third term of an AP is 12 and the seventh term is 24, then the 10th term will be?
a+2d = 12
a + 6d = 24
subtract them:
4d = 12
d = 3
back into the first: a + 6 = 12, a = 6
term(10) = a + 9d = 6 + 27 = 33
thanks
Brilliant
Well, in this case, we know that the difference between consecutive terms in the arithmetic progression (AP) is constant. So, let's calculate that difference using the third and seventh terms. The difference between the third and seventh terms is 24 - 12 = 12.
Now, to find the 10th term, we can keep adding this difference to the previous term. Starting with the third term (12), we add the difference three times to get:
12 + 12 + 12 + 12 = 48
Therefore, the 10th term of the AP will be 48.
To find the 10th term of an arithmetic progression (AP) given the third and seventh terms, we first need to find the common difference (d), and then use that common difference to find the 10th term.
An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is always the same. It can be represented as a_n = a_1 + (n-1)d, where a_n is the nth term, a_1 is the first term, n is the term number, and d is the common difference.
Given that the third term of the AP is 12, we can express it as:
a_3 = a_1 + (3-1)d
12 = a_1 + 2d ---(1)
Similarly, given that the seventh term is 24, we can express it as:
a_7 = a_1 + (7-1)d
24 = a_1 + 6d ---(2)
Now, we can use these equations (1) and (2) to solve for a_1 and d simultaneously.
Subtracting equation (1) from equation (2), we get:
24 - 12 = a_1 + 6d - a_1 - 2d
12 = 4d
Dividing both sides of the equation by 4, we find:
d = 3
Now that we have the common difference, we can substitute it back into equation (1) to find a_1:
12 = a_1 + 2(3)
12 = a_1 + 6
a_1 = 6
Finally, substituting the values of a_1 (6) and d (3) into the formula a_n = a_1 + (n-1)d, we can find the 10th term:
a_10 = 6 + (10-1)(3)
a_10 = 6 + 9(3)
a_10 = 6 + 27
a_10 = 33
Therefore, the 10th term of the arithmetic progression is 33.