In the AP (25,33,41,\ldots), find the sum from the (40)th term to the (70)th term.
Answer and explanation
Correct answer: (14167)
The sequence \(25,33,41,\ldots\) is an arithmetic progression because the difference between consecutive terms is always \(8\). Its first term is \(25\), so the \(n\)th term is \(a_n=25+(n-1)8\). The required range contains terms 40 through 70, giving \(70-40+1=31\) terms.
The 40th term is \(25+39(8)=337\), and the 70th term is \(25+69(8)=577\). The sum of a finite AP is \(\frac{\text{number of terms}}{2}(\text{first term} + \text{last term})\). Hence the required sum is \(\frac{31}{2}(337+577)=\frac{31}{2}(914)=31(457)=14167\). Therefore option B is correct; using 30 instead of 31 terms would incorrectly exclude one endpoint.
Frequently asked questions
What is the correct answer to this question?
(14167)
Why is this the correct answer?
The sequence \(25,33,41,\ldots\) is an arithmetic progression because the difference between consecutive terms is always \(8\). Its first term is \(25\), so the \(n\)th term is \(a_n=25+(n-1)8\). The required range contains terms 40 through 70, giving \(70-40+1=31\) terms.
The 40th term is \(25+39(8)=337\), and the 70th term is \(25+69(8)=577\). The sum of a finite AP is \(\frac{\text{number of terms}}{2}(\text{first term} + \text{last term})\). Hence the required sum is \(\frac{31}{2}(337+577)=\frac{31}{2}(914)=31(457)=14167\). Therefore option B is correct; using 30 instead of 31 terms would incorrectly exclude one endpoint.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Arithmetic Progressions (AP). Topic: Finding the sum of the first $n$ terms of an AP.
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