The value of \(S_n-S_{n-1}+S_{n-4}-S_{n-5}\) is 150. What is \(n\), where \(S_k=1+2+\cdots+k\)?
Answer and explanation
Correct answer: 77
The key property is \(S_k-S_{k-1}=k\), because subtracting two consecutive sums leaves only the kth term. Apply it to both pairs: \(S_n-S_{n-1}=n\), while \(S_{n-4}-S_{n-5}=n-4\). Hence the complete expression becomes \(n+(n-4)=2n-4\). Setting it equal to 150 gives \(2n-4=150\), so \(2n=154\) and \(n=77\). Therefore option C is correct. The nearby choices arise from losing the minus 4, using the wrong consecutive term, or making a one-unit arithmetic error; they do not reproduce the stated expression.
Frequently asked questions
What is the correct answer to this question?
77
Why is this the correct answer?
The key property is \(S_k-S_{k-1}=k\), because subtracting two consecutive sums leaves only the kth term. Apply it to both pairs: \(S_n-S_{n-1}=n\), while \(S_{n-4}-S_{n-5}=n-4\). Hence the complete expression becomes \(n+(n-4)=2n-4\). Setting it equal to 150 gives \(2n-4=150\), so \(2n=154\) and \(n=77\). Therefore option C is correct. The nearby choices arise from losing the minus 4, using the wrong consecutive term, or making a one-unit arithmetic error; they do not reproduce the stated expression.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Sequences and Progressions. Topic: Sum of first n natural numbers.
Student feedback
Was this question useful?
👍 0 Helpful 👎 0 Not helpful
Yes 0% No 0%
0 responsesStudent Reviews
No published reviews yet.
