The first term of an arithmetic progression is (12) and the sum of the first (10) terms is (345). What is the common difference?
Answer and explanation
Correct answer: 5
The sum of the first \(n\) terms of an AP is \(S_n=\frac{n}{2}[2a+(n-1)d]\). Here, \(345=\frac{10}{2}[2(12)+9d]=5(24+9d)\). Thus, \(24+9d=69\), so \(9d=45\) and \(d=5\). If \(d=4\), the sum would be 300, not 345. Exam tip: substitute the given \(a\), \(n\), and \(S_n\) directly into the sum formula to find \(d\).
Frequently asked questions
What is the correct answer to this question?
5
Why is this the correct answer?
The sum of the first \(n\) terms of an AP is \(S_n=\frac{n}{2}[2a+(n-1)d]\). Here, \(345=\frac{10}{2}[2(12)+9d]=5(24+9d)\). Thus, \(24+9d=69\), so \(9d=45\) and \(d=5\). If \(d=4\), the sum would be 300, not 345. Exam tip: substitute the given \(a\), \(n\), and \(S_n\) directly into the sum formula to find \(d\).
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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