in an AP the 10th term is 54 and 5th term is 4 times the 2nd term . Find the 1st term 'a' and common difference 'd'
a+9d = 54
a+4d = 4
5d=50
d = 10
a = -36
To find the 1st term (a) and the common difference (d) in an arithmetic progression (AP), we can use the given information.
We are told that the 10th term is 54, and we need to find a and d.
The formula to find the nth term (T_n) in an arithmetic progression is:
T_n = a + (n - 1)d
First, we can find the value of the 10th term (T_10) using this formula:
T_10 = a + (10 - 1)d
54 = a + 9d -- equation (1)
Next, we are given that the 5th term (T_5) is 4 times the 2nd term (T_2).
We can substitute the values into the formula to express this:
T_5 = 4 * T_2
(a + (5 - 1)d) = 4 * (a + (2 - 1)d)
(a + 4d) = 4a + 3d
3a = d -- equation (2)
Now, we have two equations with two variables (a and d). We can solve these equations simultaneously to find the values of a and d.
Substituting equation (2) into equation (1), we get:
54 = a + 9(3a)
54 = a + 27a
54 = 28a
a = 54/28
a = 27/14
So, the first term (a) is 27/14.
Substituting this value of a back into equation (2), we get:
3(27/14) = d
81/14 = d
Therefore, the common difference (d) is 81/14.