If Sₙ = 3n² + 4n, find the sum of the 21st term through the 30th term.
Answer and explanation
Correct answer: 1540
If \(S_n\) is the sum of the first \(n\) terms, the sum from the 21st through the 30th term is \(S_{30}-S_{20}\). The subtraction removes the first 20 terms and leaves exactly terms 21 to 30. Using the given rule, \(S_{30}=3(30)^2+4(30)=2700+120=2820\), while \(S_{20}=3(20)^2+4(20)=1200+80=1280\).
Therefore, the required sum is \(2820-1280=1540\). This matches option C. It is important not to use \(S_{30}-S_{21}\), because that would remove the first 21 terms and begin with the 22nd term. The endpoints are included: the calculation must contain both the 21st and the 30th terms, and subtracting \(S_{20}\) does exactly that.
Frequently asked questions
What is the correct answer to this question?
1540
Why is this the correct answer?
If \(S_n\) is the sum of the first \(n\) terms, the sum from the 21st through the 30th term is \(S_{30}-S_{20}\). The subtraction removes the first 20 terms and leaves exactly terms 21 to 30. Using the given rule, \(S_{30}=3(30)^2+4(30)=2700+120=2820\), while \(S_{20}=3(20)^2+4(20)=1200+80=1280\).
Therefore, the required sum is \(2820-1280=1540\). This matches option C. It is important not to use \(S_{30}-S_{21}\), because that would remove the first 21 terms and begin with the 22nd term. The endpoints are included: the calculation must contain both the 21st and the 30th terms, and subtracting \(S_{20}\) does exactly that.
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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