In the AP (18,25,32,\ldots), find the sum from the (30)th term to the (55)th term.
Answer and explanation
Correct answer: (8021)
The sequence is an arithmetic progression with first term 18 and common difference 7. Its nth term is \(a_n=18+(n-1)7\). The requested terms are the 30th through the 55th, so their number is \(55-30+1=26\). The first term of this block is \(a_{30}=221\), and the last is \(a_{55}=396\). The sum of consecutive AP terms equals the number of terms multiplied by the average of the first and last terms.
Hence the sum is \(26\times\frac{221+396}{2}=13\times617=8021\). The same result follows from \(S_{55}-S_{29}\), because subtracting through the 29th term leaves precisely terms 30 to 55. Therefore option A is correct. Subtracting \(S_{30}\) would incorrectly omit the 30th term, so the endpoint handling is important.
Frequently asked questions
What is the correct answer to this question?
(8021)
Why is this the correct answer?
The sequence is an arithmetic progression with first term 18 and common difference 7. Its nth term is \(a_n=18+(n-1)7\). The requested terms are the 30th through the 55th, so their number is \(55-30+1=26\). The first term of this block is \(a_{30}=221\), and the last is \(a_{55}=396\). The sum of consecutive AP terms equals the number of terms multiplied by the average of the first and last terms.
Hence the sum is \(26\times\frac{221+396}{2}=13\times617=8021\). The same result follows from \(S_{55}-S_{29}\), because subtracting through the 29th term leaves precisely terms 30 to 55. Therefore option A is correct. Subtracting \(S_{30}\) would incorrectly omit the 30th term, so the endpoint handling is important.
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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