CBSE Class 10th Exam Date:17 Feb' 26 - 17 Feb' 26
A progression is a sequence (or a set) of numbers that follow a specific pattern of repetition. We can define Arithmetic Progression as a set of numbers where the difference between any two consecutive terms remains the same throughout the series. This difference between the two terms is called the common difference of the AP. The general form of an AP is a, a + d, a + 2d, .... , where d is the common difference, and a is the first term.
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A brief summary is also provided at the end of this NCERT Solutions Class 10 Mathematics chapter 5 exercise 5.4, which will aid students in quickly memorising the full chapter as well as the formulas. NCERT Book Exercise 5.4 Class 10 Maths is an optional exercise with a total of 5 questions that may appear difficult at first and will necessitate some brainstorming.
Answer:
Given AP is
$\small 121,117,113,...,$
Here $a = 121 \ and \ d = -4$
Let's suppose the nth term of the AP is the first negative term
Then,
$a_n = a+ (n-1)d$
If the nth term is negative, then $a_n < 0$
$\Rightarrow 121+(n-1)(-4) < 0$
$\Rightarrow 125<4n$
$\Rightarrow n > \frac{125}{4}=31.25$
Therefore, the first negative term must be the 32nd term
Answer:
It is given that the sum of the third and seventh terms of an AP is and their product is $\small 8$
$a_3= a+ 2d$
$a_7= a+ 6d$
Now,
$a_3+a_7= a+ 2d+a+6d= 6$
$\Rightarrow 2a+8d = 6$
$\Rightarrow a+4d = 3 \Rightarrow a = 3-4d \ \ \ \ \ \ \ \ \ \ \ \ -(i)$
And
$a_3.a_7 = (a+2d).(a+6d)=a^2+8ad +12d^2 = 8 \ \ \ \ \ \ \ -(ii)$
Putting the value from equation (i) in equation (ii), we will get
$\Rightarrow (3-4d)^2+8(3-4d)d+12d^2= 8$
$\Rightarrow 9+16d^2-24d+24d-32d^2+12d^2=8$
$\Rightarrow 4d^2 = 1$
$\Rightarrow d = \pm \frac{1}{2}$
Now,
Case (i) $d = \frac{1}{2}$
$a= 3 - 4 \times \frac{1}{2} = 1$
Then,
$S_{16}=\frac{16}{2}\left \{ 2\times 1+(16-1)\frac{1}{2} \right \}$
$S_{16}=76$
Case (ii) $d = -\frac{1}{2}$
$a= 3 - 4 \times \left ( -\frac{1}{2} \right ) = 5$
Then,
$S_{16}=\frac{16}{2}\left \{ 2\times 1+(16-1)\left ( -\frac{1}{2} \right ) \right \}$
$S_{16}=20$
Q3. A ladder has rungs $\small 25$ cm apart. (see Fig. $\small 5.7$ ). The rungs decrease uniformly in length from $\small 45$ cm at the bottom to $\small 25$ cm at the top. If the top and the bottom rungs are $\small 2\frac{1}{2}$ m apart, what is the length of the wood required for the rungs? [Hint:Number of rungs = $\frac{250}{25}+1$]
Answer:
It is given that
The total distance between the top and bottom rung $= 2\frac{1}{2}\ m = 250cm$
Distance between any two rungs = 25 cm
Total number of rungs = $\frac{250}{25}+1= 11$
And it is also given that the bottom-most rung is of 45 cm length and the top-most is of 25 cm length. As it is given that the length of rungs
decreases uniformly, it will form an AP with $a = 25, a_{11} = 45 \ and \ n = 11$
Now, we know that
$a_{11}= a+ 10d$
$\Rightarrow 45=25+10d$
$\Rightarrow d = 2$
Now, the total length of the wood required for the rungs is equal to
$S_{11} = \frac{11}{2}\left \{ 2\times 25+(11-1)2 \right \}$
$S_{11} = \frac{11}{2}\left \{ 50+20 \right \}$
$S_{11} = \frac{11}{2}\times 70$
$S_{11} =385 \ cm$
Therefore, the total length of the wood required for the rungs is equal to 385 cm
Q4 The houses of a row are numbered consecutively from $\ small1$ to $\ small49$. Show that there is a value of $\small x$ such that the sum of the numbers of the houses preceding the house numbered $\small x$ is equal to the sum of the numbers of the houses following it. Find the value of $\ small x$.
Answer:
It is given that the sum of the numbers of the houses preceding the house numbered $\small x$ is equal to the sum of the numbers of the
houses following it
And 1,2,3,.....,49 form an AP with a = 1 and d = 1
Now, we know that
$S_n= \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
Suppose there exists an n term such that ( n < 49)
Now, according to the given conditions
The sum of the first n - 1 terms of AP = The sum of the terms following the nth term
Sum of the first n - 1 terms of AP = Sum of the whole AP - Sum of the first m terms of AP
i.e.
$S_{n-1}=S_{49}-S_n$
$\frac{n-1}{2}\left \{ 2a+((n-1)-1)d \right \}=\frac{49}{2}\left \{ 2a+(49-1)d \right \}-\frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\frac{n-1}{2}\left \{ 2+(n-2) \right \}=\frac{49}{2}\left \{ 2+48 \right \}-\frac{n}{2}\left \{ 2+(n-1) \right \}$
$\frac{n-1}{2}\left \{ n\right \}=\frac{49}{2}\left \{ 50 \right \}-\frac{n}{2}\left \{ n+1 \right \}$
$\frac{n^2}{2}-\frac{n}{2}=1225-\frac{n^2}{2}-\frac{n}{2}$
$n^2 = 1225$
$n = \pm 35$
Given House numbers are not negative,e so we reject n = -35
Therefore, the sum of no of houses preceding house no 35 is equal to the sum of no of houses following house number 35.
Answer:
It is given that
football ground comprises of $\small 15$ steps each of which is $\small 50$ m long and Each step has a rise of $\small \frac{1}{4}\: m$
and a tread of $\small \frac{1}{2}\: m$
Now,
The volume required to make the first step = $\frac{1}{4}\times \frac{1}{2}\times 50 = 6.25 \ m^3$
Similarly,
The volume required to make 2nd step = $\left ( \frac{1}{4}+\frac{1}{4}\right )\times \frac{1}{2}\times 50=\frac{1}{2}\times \frac{1}{2}\times 50 = 12.5 \ m^3$
And
The volume required to make 3rd step = $\left ( \frac{1}{4}+\frac{1}{4}+\frac{1}{4}\right )\times \frac{1}{2}\times 50=\frac{3}{4}\times \frac{1}{2}\times 50 = 18.75 \ m^3$
And so on
We can see that this is an AP with $a= 6.25 \ and \ d = 6.25$
Now, the total volume of concrete required to build the terrace of 15 such steps is
$S_{15} =\frac{15}{2}\left \{ 2 \times 6.25 +(15-1)6.25 \right \}$
$S_{15} =\frac{15}{2}\left \{ 12.5 +87.5\right \}$
$S_{15} =\frac{15}{2}\times 100$
$S_{15} =15\times 50 = 750$
Therefore, the total volume of concrete required to build the terrace of 15 such steps is $750 \ m^3$
Also Read
Also see-
Students must check the NCERT solutions for class 10 of Mathematics and Science Subjects.
Students must check the NCERT Exemplar solutions for class 10 of the Mathematics and Science Subjects.
Frequently Asked Questions (FAQs)
We can define Arithmetic Progression as a set of collective numbers where the difference between any two consecutive terms remains the same throughout the series. This difference between the two terms is called the common difference of the AP.
The differences between any two consecutive terms are the same which is known as the common difference of the arithmetic progression.
Yes, it the Arithmetic progression is a decreasing one. For example:
15, 12, 9, 6, 3
Common difference =7-4=3
nth term=a+(n-1)d
Where 'a' is the first term and 'd' is the common difference.
First ten natural numbers are 1, 2, 3, .........,10
This is ap with a common difference of 1.
Sum =0.5n(first term+ last term)
=0.5 x 10(1+10)=5 x11=55
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