the frst term of AP is 4 and the last term is 79.given that the sum of the AP is 996, find the number of terms and the commmon difference.

To find the number of terms and the common difference of an arithmetic progression (AP), we can use the following formulas:

1) The nth term of an AP: An = a + (n - 1)d

2) The sum of the first n terms of an AP: Sn = (n/2) * (2a + (n-1)d)

Given:
First term (a) = 4
Last term (An) = 79
Sum of the AP (Sn) = 996

Using the first formula, we can substitute the values of a, An, and n to find the common difference (d):

79 = 4 + (n - 1)d

Simplifying,
75 = (n - 1)d --------(1)

Now, using the second formula, we can substitute the values of a, Sn, and n, to get an equation involving n:

996 = (n/2)(2 * 4 + (n - 1)d)
996 = (n/2)(8 + (n - 1)d)
Multiplying both sides by 2 to eliminate the fraction:
1992 = n(8 + (n - 1)d)
1992 = 8n + (n^2 - n)d
1992 = 8n + nd - nd + nd
1992 = n^2d + 8n ------------(2)

Now, we have two equations (1) and (2) with two variables (n and d). We can solve these equations simultaneously using algebra or substitution to find the values of n and d.

To find the number of terms and the common difference of an arithmetic progression (AP) given its first term, last term, and sum, you can use the following formulas:

1. The formula for the sum of an AP is:
S = (n/2)(a + l), where S is the sum of the AP, n is the number of terms, a is the first term, and l is the last term.

2. The formula for the nth term of an AP is:
an = a + (n - 1)d, where an is the nth term, a is the first term, n is the number of terms, and d is the common difference.

Let's solve the problem step by step:

Step 1: Given information
- The first term (a) is 4.
- The last term (l) is 79.
- The sum of the AP (S) is 996.

Step 2: Finding the number of terms (n)
We know that S = (n/2)(a + l)

Plugging in the values:
996 = (n/2)(4 + 79)

Simplifying the equation:
996 = (n/2)(83)
996 = (83n/2)

Multiplying both sides by 2 to get rid of the fraction:
1992 = 83n

Dividing both sides by 83:
n = 24

Hence, there are 24 terms in the arithmetic progression.

Step 3: Finding the common difference (d)
We can now calculate the common difference using the first term (a), last term (l), and the number of terms (n).
We know that l = a + (n - 1)d

Plugging in the values:
79 = 4 + (24 - 1)d

Simplifying the equation:
79 = 4 + 23d

Subtracting 4 from both sides:
75 = 23d

Dividing both sides by 23:
d = 3

Hence, the common difference is 3.

Therefore, the number of terms in the AP is 24 and the common difference is 3.

a=4

Sn = n/2 (8+(n-1)d) = 996
An = 4+(n-1)d = 79

n = 24
d = 75/23