For which (r) will (r-1, 2r+2, 4r+7) be in an arithmetic progression, and what will be the common difference?
Answer and explanation
Correct answer: \(r=-2,\ d=1\)
In an arithmetic progression, the differences between consecutive terms must be equal. Here, the difference between the second and first terms is \((2r+2)-(r-1)=r+3\), while that between the third and second terms is \((4r+7)-(2r+2)=2r+5\). Thus, \(r+3=2r+5\), giving \(r=-2\). The terms then become \(-3,-2,-1\), so the common difference is \(d=1\). For \(r=0\), the two differences are 3 and 5, so that option is not correct. Exam tip: for three AP terms, also check whether \(2\times\) the middle term equals the sum of the first and third terms.
Frequently asked questions
What is the correct answer to this question?
\(r=-2,\ d=1\)
Why is this the correct answer?
In an arithmetic progression, the differences between consecutive terms must be equal. Here, the difference between the second and first terms is \((2r+2)-(r-1)=r+3\), while that between the third and second terms is \((4r+7)-(2r+2)=2r+5\). Thus, \(r+3=2r+5\), giving \(r=-2\). The terms then become \(-3,-2,-1\), so the common difference is \(d=1\). For \(r=0\), the two differences are 3 and 5, so that option is not correct. Exam tip: for three AP terms, also check whether \(2\times\) the middle term equals the sum of the first and third terms.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Arithmetic Progressions (AP). Topic: Introduction to APs and common difference..
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