(a) The fourth term of an A.P is 37 and 6th term is 12 more than the fourth term . Find the first and seventh terms. (b) If \(P = {1, 2, 3, 4}\) and \(Q = {...

Answer Details

(a) Arithmetic Progression

Let the first term be \(a\) and common difference \(d\). The \(n\)th term is \(a+(n-1)d\).

Fourth term: \(a + 3d = 37\).

The 6th term is 12 more than the 4th term: \((a+5d) = 37 + 12 = 49\), so \(a + 5d = 49\).

Subtract the equations:

\[(a+5d)-(a+3d) = 49-37 \;\Rightarrow\; 2d = 12 \;\Rightarrow\; d = 6\]

Then \(a = 37 - 3(6) = 19\).

First term \(= 19\). Seventh term:

\[a + 6d = 19 + 6(6) = 55\]

Seventh term \(= 55\).

(b) Sets with \(P=\{1,2,3,4\}\) and \(Q=\{3,5,6\}\).

(i) \(P\cap Q = \{3\}\)

(ii) \(P\cup Q = \{1,2,3,4,5,6\}\)

(iii) \((P\cap Q)\cup Q = \{3\}\cup\{3,5,6\} = \{3,5,6\}\)

(iv) \((P\cap Q)\cup P = \{3\}\cup\{1,2,3,4\} = \{1,2,3,4\}\)