The second, fourth and sixth terms of an…

Question 1 Report

The second, fourth and sixth terms of an Arithmetic Progression (AP.) are x - 1, x + 1 and 7 respectively. Find the:

(a) common difference;

(b) first term;

Answer

Answer Details

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

Second term: \(a+d=x-1\)
Fourth term: \(a+3d=x+1\)
Sixth term: \(a+5d=7\)

(a) Common difference. Subtract the second-term equation from the fourth-term equation:

\[(a+3d)-(a+d)=(x+1)-(x-1)\Rightarrow 2d=2\Rightarrow d=1\]

Common difference \(d=1\).

(c) Value of x. Subtract the fourth-term equation from the sixth-term equation:

\[(a+5d)-(a+3d)=7-(x+1)\Rightarrow 2d=6-x\]

With \(d=1\): \(2=6-x\Rightarrow x=4\).

(b) First term. From \(a+d=x-1\): \(a+1=4-1=3\Rightarrow a=2\).

First term \(=2\), common difference \(=1\), \(x=4\). (Check: terms are \(2,3,4,5,6,7,\ldots\); 2nd\(=3=x-1\), 4th\(=5=x+1\), 6th\(=7\).)

The second, fourth and sixth terms of an Arithmetic Progression (AP.) are x - 1, x + 1 and 7 respectively. Find the: (a) common difference; (b) first term; ...