the 54th term and 4th term of an A.P are -61 and 64 respectively.Find the common difference and 23rd tem
the 54th term is 50 differences from the 4th term
To find the common difference and the 23rd term of an arithmetic progression (A.P.), we can use the formulas:
nth term (An) = a + (n-1)d
Given:
a1 (first term) = 4
a54 (54th term) = -61
Let's find the common difference (d) first:
a54 = a1 + (54-1)d
-61 = 4 + 53d
-61 - 4 = 53d
-65 = 53d
d = -65/53
Now, let's find the 23rd term (a23):
a23 = a1 + (23-1)d
a23 = 4 + 22(-65/53)
a23 = 4 - 1430/53
a23 = -336/53
Therefore, the common difference is -65/53 and the 23rd term is -336/53.
To find the common difference (d) and the 23rd term (a23) of an arithmetic progression (A.P.), we can use the formulas below:
nth term of an A.P.: an = a1 + (n - 1)d (where an is the nth term, a1 is the first term, and d is the common difference)
From the given information, we have:
a54 = -61
a4 = 64
Using the formula for the nth term, we can substitute the given values to form two equations:
a54 = a1 + (54 - 1)d ... (1)
a4 = a1 + (4 - 1)d ... (2)
We can rearrange equation (1) to solve for a1:
a1 = a54 - (54 - 1)d
a1 = -61 - 53d ... (3)
Now, let's substitute the value of a1 from equation (3) into equation (2):
64 = (-61 - 53d) + 3d
64 = -61 - 50d
125 = -50d
d = -125/50
d = -2.5
Therefore, the common difference is -2.5.
To find the 23rd term (a23), we can use equation (1) by substituting the values we have now:
a23 = a1 + (23 - 1)d
a23 = (-61 - 53(-2.5)) + 22(-2.5)
a23 = -61 + 132.5 - 55
a23 = 16.5
Hence, the 23rd term of the arithmetic progression is 16.5.