Answer :
The first term a of the A.P. is 1, and the common difference d is 4.
Therefore, the A.P. is 1, 5, 9, 13, 17, 21, 25, ...
To find the first term and the common difference of the arithmetic progression (A.P.), let's denote the first term as a and the common difference as d .
The sum of the first seven terms of an A.P. is given by the formula:
[tex]\[ S_n = \frac{n}{2}[2a + (n - 1)d] \][/tex]
where [tex]\( S_n \)[/tex] is the sum of the first n terms.
Given that the sum of the first seven terms is 182, we can write:
[tex]\[ 182 = \frac{7}{2}[2a + (7 - 1)d] \]\[ 182 = 7[2a + 6d] \]\[ 26 = 2a + 6d \]\[ 13 = a + 3d \][/tex]
Given that the 4th and 17th terms are in the ratio 1:5, we can express this as:
[tex]\[ \frac{a + 3d}{a + 16d} = \frac{1}{5} \][/tex]
Cross-multiplying:
[tex]\[ 5(a + 3d) = a + 16d \]\[ 5a + 15d = a + 16d \]\[ 4a = d \][/tex]
Now, we have two equations (equations 1 and 2) which can be solved simultaneously to find the values of a and d .
From equation (2), we can express d in terms of a :
[tex]\[ d = \frac{4a}{1} \]\[ d = 4a \][/tex]
Substituting this value of d into equation (1), we get:
[tex]\[ 13 = a + 3(4a) \]\[ 13 = a + 12a \]\[ 13 = 13a \]\[ a = 1 \][/tex]
Now, substituting a = 1 into equation (2), we get:
[tex]\[ d = 4(1) \]\[ d = 4 \][/tex]
So, the first term a of the A.P. is 1, and the common difference d is 4.
Therefore, the A.P. is 1, 5, 9, 13, 17, 21, 25, ...
