The sum of the 2nd and 5th terms of an arithmetic progression (AP) is 42. If the difference between the 6th and 3rd term is 12, find the (i) Common differen...
Question 1 Report
The sum of the 2nd and 5th terms of an arithmetic progression (AP) is 42. If the difference between the 6th and 3rd term is 12, find the
(i) Common difference
(ii) first term
(iii) 20th term.
Answer Details
Let the first term be \(a\) and common difference \(d.\) The \(n\)th term is \(T_n=a+(n-1)d.\)
Set up the equations. Sum of 2nd and 5th terms is 42:
\[(a+d)+(a+4d)=42\Rightarrow 2a+5d=42.\]
Difference between 6th and 3rd terms is 12:
\[(a+5d)-(a+2d)=12\Rightarrow 3d=12.\]
(i) Common difference. \(3d=12\Rightarrow d=4.\)
(ii) First term. Substitute into \(2a+5d=42:\)
\[2a+5(4)=42\Rightarrow 2a=22\Rightarrow a=11.\]
(iii) 20th term.
\[T_{20}=a+19d=11+19(4)=11+76=87.\]
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