the third term of A.p is 12 while the seventh term is 72 find (1)common differences
(2) first term
(3) find the sixth term
a+2d = 12
a+6d = 72
I think you can now easily find a and d and find other terms, right?
To find the common difference in an arithmetic progression (AP), you subtract any term from its preceding term. In this case, given the third term as 12 and the seventh term as 72, we can find the common difference.
(1) To find the common difference:
The third term is 12, and we count back two terms to get the first term.
12 - (2 * d) = a
where d is the common difference, and a is the first term.
In this case, substituting the values, we get:
12 - (2 * d) = a
Now we can use the seventh term, 72, to get another equation:
a + (6 - 1) * d = 72
where 6 is the position of the seventh term in the AP.
Substituting the value of "a" from the first equation:
(12 - 2d) + (6 - 1) * d = 72
Simplifying the equation:
12 - 2d + 5d = 72
-2d + 5d = 72 - 12
3d = 60
d = 60/3
d = 20
Therefore, the common difference is 20.
(2) To find the first term (a):
Using the first equation from earlier:
12 - (2 * 20) = a
12 - 40 = a
a = -28
Therefore, the first term is -28.
(3) To find the sixth term:
We can use the formula for the nth term of an arithmetic progression:
tn = a + (n - 1) * d
where tn is the nth term, a is the first term, n is the position of the term in the AP, and d is the common difference.
Substituting the values:
t6 = -28 + (6 - 1) * 20
t6 = -28 + 5 * 20
t6 = -28 + 100
t6 = 72
Therefore, the sixth term is 72.