A child arranges marbles in rows so that the first row has 1 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?

Question

Homi · Accepted Answer

Understand the Question

The number of marbles in each row forms a sequence of consecutive positive integers: $1, 2, 3, \dots, 25$ .
To find the total number of marbles used, calculate the sum of the first $n = 25$ natural numbers using the formula: $S_n = \dfrac{n(n+1)}{2}$

Step 1 · Identify the Arithmetic Progression

The number of marbles in each row forms an arithmetic progression (AP): Diagram 1

First term ( $a$ ) $= 1$
Common difference ( $d$ ) $= 1$
Number of rows ( $n$ ) $= 25$

Step 2 · Calculate the Total Sum

Using the formula for the sum of the first $n$ natural numbers: $S_n = \dfrac{n(n+1)}{2}$

Substitute $n = 25$ :

\begin{aligned} S_{25} &= \dfrac{25(25+1)}{2} \\[0.6em] &= \dfrac{25 \times 26}{2} \\[0.6em] &= 25 \times 13 \\[0.6em] &= 325 \end{aligned}

Answer

325 \text{ marbles}

Common Mistakes

Arithmetic Error: Forgetting to divide by $2$ in the formula $\dfrac{n(n+1)}{2}$ , resulting in $650$ instead of $325$ .
Incorrect Number of Terms: Miscounting the number of rows as $n = 24$ or $n = 26$ instead of $n = 25$ .

Question 7

More questions in Exercise 8.2