QUESTION 7
Easy
A child arranges marbles in rows so that the first row has one marble, the second has 2 marbles, the third has 3, and so on up to 25 rows. How many marbles does the child use in all?
SOLUTION
1
Write the AP
The arrangement of the marbles forms an AP: 1, 2, 3, ..., 25
Here,
first term, \(a = 1\)
last term, \(l = 25\)
number of terms, \(n = 25\)
Here,
first term, \(a = 1\)
last term, \(l = 25\)
number of terms, \(n = 25\)
2
Use the sum formula
\(S_n = \frac{n}{2}(a + l)\)
⇒ \(S_{25} = \frac{25}{2}(1 + 25)\)
⇒ \(S_{25} = \frac{25}{2}26\)
⇒ \(S_{25} = 25 × 13\)
⇒ \(S_{25} = 325\)
So, total marbles used = 325
⇒ \(S_{25} = \frac{25}{2}(1 + 25)\)
⇒ \(S_{25} = \frac{25}{2}26\)
⇒ \(S_{25} = 25 × 13\)
⇒ \(S_{25} = 325\)
So, total marbles used = 325
🏆
Final Answer : The child uses 325 marbles in all.
Concept Note
The sum of an AP is given by the formula
\(S_n = \frac{n}{2}(a + l)\)
where
\(n = \)number of terms
\(a = \)first term
\(l = \)last term