On a stage, the rth row has 6r lamps. How many lamps will there be in 45 rows?
Answer and explanation
Correct answer: 6210
The number of lamps increases in direct proportion to the row number: row 1 has 6 lamps, row 2 has 12, and row r has 6r. Therefore, the total for 45 rows is 6 times the sum of the first 45 natural numbers. This is an arithmetic progression whose first term is 6, last term is 270, and number of terms is 45.
Using the sum formula, the total is \(6(1+2+\cdots+45)=6\times\frac{45(45+1)}{2}\). Thus it is \(6\times\frac{45\times46}{2}=6\times1035=6210\). Hence option B follows. The incomplete supplied explanation points toward the right method but does not finish the calculation; the answer itself is consistent.
Frequently asked questions
What is the correct answer to this question?
6210
Why is this the correct answer?
The number of lamps increases in direct proportion to the row number: row 1 has 6 lamps, row 2 has 12, and row r has 6r. Therefore, the total for 45 rows is 6 times the sum of the first 45 natural numbers. This is an arithmetic progression whose first term is 6, last term is 270, and number of terms is 45.
Using the sum formula, the total is \(6(1+2+\cdots+45)=6\times\frac{45(45+1)}{2}\). Thus it is \(6\times\frac{45\times46}{2}=6\times1035=6210\). Hence option B follows. The incomplete supplied explanation points toward the right method but does not finish the calculation; the answer itself is consistent.
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.
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