Ex 5.4 Class 10 Maths | NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions Ex 5.4
NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions Ex 5.4
Question 1.
Which term of the AP: 121, 117. 113, ….., is its first negative term?
Solution:
Given, a = 121, d = -4
We use the formula:
an = a + (n – 1)d
Set an < 0 to find the first negative term:
121 + (n – 1)(-4) < 0
⇒ 121 – 4(n – 1) < 0
⇒ 121 – 4n + 4 < 0
⇒ 125 – 4n < 0
⇒ 4n > 125
⇒ n > 31.25
So, the first negative term is the 32nd term.
Answer: 32nd term
Ex 5.4 Class 10 Maths
Question 2.
The sum of the third term and the seventh term of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP?
Solution:
Ex 5.4 Class 10 Maths
Question 3.
A ladder has rungs 25 cm apart. The rungs decrease uniformly in length from 45 cm at the bottom to 25 at the top. If the top and the bottom rungs are [Math Processing Error] apart, what is the length of the wood required for the rungs?
Solution:
Ex 5.4 Class 10 Maths
Question 4.
The houses of a row are numbered consecutively from 1 to 49. Show that there is value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x.
Solution:
Ex 5.4 Class 10 Maths | NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions Ex 5.4
Question 5.
A small terrace at a football ground comprises of 15 steps each of which is 50 m long and built of solid concrete. Each step has a rise of [Math Processing Error] and a tread of [Math Processing Error]. Calculate the total volume of concrete required to build the terrace.
From the figure, we see that:
1st step is 1/2 m wide,
2nd step is 1 m wide,
3rd step is 3/2 m wide.
Therefore, the width of each step is increasing by 1/2 m each time, whereas their height (1/4 m) and length (50 m) remain the same.
So, the widths of the steps form an AP:
1/2, 1, 3/2, 2, …
Now, the volume of concrete in each step:
Volume of 1st step = 1/4 × 1/2 × 50 = 25/4 m³
Volume of 2nd step = 1/4 × 1 × 50 = 25/2 m³
Volume of 3rd step = 1/4 × 3/2 × 50 = 75/4 m³
So, the volumes of concrete form an AP:
25/4, 25/2, 75/4, …
Let’s find the sum of volumes for 15 steps.
This is the sum of the first 15 terms of an AP.
First term (a) = 25/4
Common difference (d) = 25/2 – 25/4 = 25/4
Number of terms (n) = 15
Use the AP sum formula:
Sₙ = n/2 × [2a + (n – 1)d]
S₁₅ = 15/2 × [2 × (25/4) + (15 – 1) × (25/4)]
= 15/2 × [(50/4) + (350/4)]
= 15/2 × (400/4)
= 15/2 × 100
= 750
Answer: The volume of concrete required to build the terrace is 750 m³
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