If (S_n=3n^2+2n), find the sum from the (51)st term to the (80)th term.
Answer and explanation
Correct answer: (11760)
The expression \(S_n=3n^2+2n\) gives the sum of the first \(n\) terms. To find the sum from the 51st through the 80th term, subtract the sum of the first 50 terms from the sum of the first 80 terms. This leaves exactly the required terms, since all terms before the 51st are removed.
Compute \(S_{80}=3(80)^2+2(80)=3(6400)+160=19360\). Also, \(S_{50}=3(50)^2+2(50)=3(2500)+100=7600\). Therefore the required sum is \(S_{80}-S_{50}=19360-7600=11760\). Hence option B is correct. The subtraction must use \(S_{50}\), not \(S_{51}\), because the 51st term must remain in the answer.
Frequently asked questions
What is the correct answer to this question?
(11760)
Why is this the correct answer?
The expression \(S_n=3n^2+2n\) gives the sum of the first \(n\) terms. To find the sum from the 51st through the 80th term, subtract the sum of the first 50 terms from the sum of the first 80 terms. This leaves exactly the required terms, since all terms before the 51st are removed.
Compute \(S_{80}=3(80)^2+2(80)=3(6400)+160=19360\). Also, \(S_{50}=3(50)^2+2(50)=3(2500)+100=7600\). Therefore the required sum is \(S_{80}-S_{50}=19360-7600=11760\). Hence option B is correct. The subtraction must use \(S_{50}\), not \(S_{51}\), because the 51st term must remain in the answer.
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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