Arithmetic Progressions | Exercise 5.4

Question 3

A ladder has rungs 25 cm apart. (see Fig. 5.7). The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and the bottom rungs are $2\dfrac{1}{2}\text{ m}$ apart, what is the length of the wood required for the rungs?

[Hint : Number of rungs = $\dfrac{250}{25} + 1$ ]

Solution

Understand the Question

The total distance between the top and bottom rungs is $2\dfrac{1}{2}\text{ m} = 250\text{ cm}$ , and the rungs are spaced $25\text{ cm}$ apart.
The number of rungs is given by $\dfrac{\text{Total distance}}{\text{Distance between rungs}} + 1$ .
Since the lengths of the rungs decrease uniformly, they form an Arithmetic Progression (AP) where:
- First term ( $a$ ) = length of top rung = $25\text{ cm}$
- Last term ( $l$ or $a_n$ ) = length of bottom rung = $45\text{ cm}$
The total length of wood required is the sum of the lengths of all rungs ( $S_n$ ).

Step 1 · Calculate the Number of Rungs

Total distance between the top and bottom rungs: $\text{Total distance} = 2\dfrac{1}{2}\text{ m} = 2.5 \times 100 = 250\text{ cm}$

Distance between successive rungs $= 25\text{ cm}$ . Diagram 1

$\text{Number of gaps} = \dfrac{250}{25} = 10$

\begin{aligned} \text{Number of rungs } (n) &= \text{Number of gaps} + 1 \\[0.6em] &= 10 + 1 \\[0.6em] &= 11 \end{aligned}

Step 2 · Define the Arithmetic Progression

Let the lengths of the rungs form an AP from top to bottom:

First term ( $a$ ) $= 25\text{ cm}$
$11^{\text{th}}$ term ( $a_{11}$ ) $= 45\text{ cm}$
Number of terms ( $n$ ) $= 11$

Using $a_n = a + (n - 1)d$

\begin{aligned} a_{11} &= a + (11 - 1)d \\[0.6em] 45 &= 25 + 10d \\[0.6em] 10d &= 20 \\[0.6em] d &= 2\text{ cm} \end{aligned}

Step 3 · Calculate the Total Length of Wood

Using the sum formula $S_n = \dfrac{n}{2}(2a + (n - 1)d)$ with $n = 11$ , $a = 25$ , and $d = 2$

\begin{aligned} S_{11} &= \dfrac{11}{2}(2(25) + (11 - 1)2) \\[0.6em] &= \dfrac{11}{2}(50 + 20) \\[0.6em] &= \dfrac{11}{2}(70) \\[0.6em] &= 11 \times 35 \\[0.6em] &= 385\text{ cm} \end{aligned}

Answer

$385\text{ cm}$

Common Mistakes

Fencepost Error: Forgetting to add $1$ when finding the number of rungs (dividing $\dfrac{250}{25} = 10$ gives the number of spaces between rungs, so the number of rungs is $10 + 1 = 11$ ).
Unit Mismatch: Not converting $2\dfrac{1}{2}\text{ m}$ to $\text{cm}$ before dividing by $25\text{ cm}$ .

Question 3

More questions in Exercise 5.4