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

We need to find the total number of marbles by summing the marbles in each row.

Step 1 — Understand the pattern

Let's look at the number of marbles in each row. The first row has 1 marble. The second row has 2 marbles. The third row has 3 marbles. This pattern continues up to 25 rows. So, we need to sum the numbers from 1 to 25. This is an arithmetic progression. The first term $a$ is 1. The common difference $d$ is 1. The number of terms $n$ is 25.

Diagram 1

Step 2 — Calculate the total sum

We can use the formula for the sum of the first $n$ natural numbers. The formula is $S_n = \frac{n(n+1)}{2}$ . Here, $n$ is 25. Let's substitute $n = 25$ into the formula.

$S_{25} = \frac{25(25+1)}{2}$

$= \frac{25 \times 26}{2}$

$= 25 \times 13$

$\boxed{325 \text{ marbles}}$

Answer

The child uses 325 marbles in all.

Question 7

Answer

More questions in Exercise 8.2