find the common difference of A.P whose first term is 5 and the sum of its first 4 terms is half the sum of next 4 terms

Let the common difference be d.

The sum of the first four terms is
5 + (5+d) + (5+d+d) + (5+d+d+d)
= 20 + 6d
The sum of the next 4 terms is
5+4d + 5+5d + 5+6d + 5+7d = 20 + 22d

You have been told that
20 +6d = (1/2)(20 + 22d)= 10 + 11d
Therefore
5d = 10
d = 2

Check: does
5+7+9+11 = (1/2)(13+15+17+19) ?
32 = (1/2)(64)

2. Find A.P Which fifth term is 5 and common difference is -3.

ask your teacher

Given 5th term is 5 and common difference is -3

a5 = a +4d
5 = a -12
a = 17
So the AP is 17 , 14 , 11

To find the common difference of an arithmetic progression (A.P.), we need to use the given information about the sum of terms. Let's break down the problem step by step:

Step 1: Recall the formula for the sum of an A.P.
The sum of the first 'n' terms of an arithmetic progression can be calculated using the formula: Sn = (n/2) * (2a + (n-1)d), where 'a' represents the first term, 'd' represents the common difference, and 'n' represents the number of terms.

Step 2: Use the formula to find the sum of the first 4 terms.
Given that the sum of the first 4 terms is half the sum of the next 4 terms, we can write the equation:
(4/2) * (2a + 3d) = (1/2) * [(8/2) * (2a + 7d)]

Simplifying both sides of the equation, we get:
2(2a + 3d) = (1/2)(4(2a + 7d))
4a + 6d = 2a + 7d
2a = d

Step 3: Determine the common difference.
From the equation, we can conclude that the common difference 'd' equals twice the first term 'a', i.e., d = 2a.

Therefore, the common difference of the given A.P. is twice the first term.