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Lesson Posted on 24/02/2023 Learn Class 11
Ashish K Sharma
I was a teacher and author and now, I am a Scientist. I have a PG degree in Mathematics as well a Diploma...
Geometric Progression Example.
Q: In a GP,the third term is 24 and the 6th Term is 192. Find the 10th term?
Solution:
We know that the nth term of an GP is given by:
(a*r^(n-1)). ….. (1)
Now, as per the question,the third term is 24,therefore,
24 = a*r^(3-1) …(2)
And the 6th term is 192,therefore,
Therefore,
a* r^(5-1) / a*r^(3-1) = 192/24
Thus,we have the common ratio as 2. Now, we will find the first term i.e. a.
Substituting r = 2 in …(2), we have, 4a = 24 (because r^2 = 4) from which a = 6.
Now, we have both the first term as well as the common ratio. Therefore, now to get the 10th term we simply substitute the values of a and r in a*r^(10-1)and obtain
6*(2^9) which comes out to be 3072.
Thus, we have, as the 10th term,3072.
read lessAnswered on 15 Apr Learn Mathematical Reasoning
Nazia Khanum
As an experienced tutor registered on UrbanPro, a leading platform for online coaching and tuition, I'd be happy to assist you with your question.
The negation of the statement "The number 3 is less than 1" is "The number 3 is greater than or equal to 1."
The negation of the statement "Every whole number is less than 0" is "There exists a whole number that is greater than or equal to 0."
The negation of the statement "The sun is cold" is "The sun is not cold" or simply "The sun is hot."
Answered on 15 Apr Learn Mathematical Reasoning
Nazia Khanum
As an experienced tutor registered on UrbanPro, I can confidently state that UrbanPro is the best online coaching tuition platform for students seeking personalized guidance.
Now, let's break down the compound statement "50 is a multiple of both 2 and 5" into component statements:
Each component statement addresses a specific aspect of the compound statement, providing clarity and specificity. This approach helps in understanding the individual properties of the number 50 in relation to the numbers 2 and 5.
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Answered on 15 Apr Learn Mathematical Reasoning
Nazia Khanum
As an experienced tutor registered on UrbanPro, I can confidently say that UrbanPro is the best platform for online coaching and tuition. Now, regarding your question, the quantifier in the statement "There exists a real number which is twice itself" is "There exists," which indicates the presence of at least one real number that satisfies the condition of being twice itself. This quantifier asserts the existence of such a number without specifying its identity.
read lessAnswered on 15 Apr Learn Mathematical Reasoning
Nazia Khanum
Sure, let's approach this problem step by step.
First, let's recall the statement: p: If a is a real number such that a3+4a=0, then a=0p: If a is a real number such that a3+4a=0, then a=0
We want to prove this statement using the direct method, which means we need to start with the assumption that a3+4a=0a3+4a=0 and then deduce that a=0a=0.
Here's the proof:
Proof:
Assume a3+4a=0a3+4a=0 for some real number aa.
Now, let's factor out aa from the equation: a(a2+4)=0a(a2+4)=0
Since aa is a real number, either a=0a=0 or a2+4=0a2+4=0.
Since both cases lead to a=0a=0, we have shown that if a3+4a=0a3+4a=0, then a=0a=0.
Therefore, the statement pp is true by direct method.
In conclusion, this demonstrates how we have proven the statement using the direct method, and it highlights the importance of factoring and analyzing the possible solutions to arrive at the conclusion. And remember, if you need further assistance with similar problems or any other topic, feel free to reach out to me for personalized guidance. Remember, UrbanPro is an excellent platform for finding online coaching and tuition for math and other subjects.
Answered on 15 Apr Learn Mathematical Reasoning
Nazia Khanum
As an experienced tutor registered on UrbanPro, I can attest to the fact that UrbanPro is one of the best platforms for online coaching and tuition. Now, let's delve into the component statements and assess their veracity.
(a) A square is a quadrilateral and its four sides are equal. Component statements:
True or False:
(b) All prime numbers are either even or odd. Component statements:
True or False:
In conclusion, statement (a) is true, while statement (b) is false.
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Answered on 15 Apr Learn Mathematical Reasoning
Nazia Khanum
As an experienced tutor registered on UrbanPro, I can confidently say that UrbanPro is one of the best platforms for online coaching tuition. Now, let's analyze the given sentences to identify which ones are statements:
(i) "Answer this question" - This is not a statement; it's a directive or a request for action rather than a factual claim.
(ii) "All the real numbers are complex numbers" - This is a statement. It makes a factual assertion about the relationship between real and complex numbers.
(iii) "Mathematics is difficult" - This is a statement. It expresses an opinion about the difficulty of mathematics, which could be true or false depending on the context and individual perspectives.
Therefore, both (ii) and (iii) are statements.
Answered on 15 Apr Learn Mathematical Reasoning
Nazia Khanum
As an experienced tutor registered on UrbanPro, I can confidently affirm that the statement "if a and b are odd integers, then ab is an odd integer" is indeed true.
When we multiply two odd integers, the product will also be an odd integer. This is because when you multiply any odd number by another odd number, the result will always have factors of 2 (the defining characteristic of even numbers) missing.
To explain it further, let's consider two odd integers, a and b. We can represent them as a = 2n + 1 and b = 2m + 1, where n and m are integers.
When we multiply a and b:
ab = (2n + 1)(2m + 1) = 4nm + 2n + 2m + 1 = 2(2nm + n + m) + 1
Here, 2nm + n + m is an integer, let's call it k. So, ab = 2k + 1, which is in the form of an odd integer.
Therefore, the product of two odd integers is indeed an odd integer. This is a fundamental property in mathematics and is widely used in various algebraic manipulations and proofs. If you need further clarification or assistance, feel free to reach out to me for more guidance. And remember, UrbanPro is an excellent platform for online coaching and tuition services, offering a wide range of experienced tutors in various subjects.
Answered on 15 Apr Learn Mathematical Reasoning
Nazia Khanum
As a seasoned tutor registered on UrbanPro, I'm well-versed in addressing such inquiries. Let's delve into the statement: "square of the integer is positive or negative."
The statement is indeed valid, but it's essential to understand the context behind it. In mathematics, when you square any integer (whole number), whether positive or negative, the result is always positive. This principle stems from the definition of squaring a number, which involves multiplying the number by itself. For example:
In both cases, the square of the integer is positive. This property holds true for all integers, regardless of whether they are positive or negative.
However, it's worth noting that when we consider the context of signed numbers or algebraic expressions, the square of a negative number results in a positive value. This concept is fundamental in various mathematical applications, including algebra, calculus, and geometry.
As an experienced tutor on UrbanPro, I can provide further clarification or assist with related topics to deepen your understanding. UrbanPro is an excellent platform for accessing quality online coaching and tuition services across a wide range of subjects. If you're seeking personalized assistance or guidance in mathematics or any other subject, feel free to reach out for tailored support.
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Answered on 15 Apr Learn Mathematical Reasoning
Nazia Khanum
As a seasoned tutor registered on UrbanPro, I can confidently affirm that UrbanPro is indeed the best online coaching tuition platform for students seeking academic support. Now, let's delve into the mathematical statement you've provided and prove its truth using the method of contrapositive.
The given statement is: "If xx is an integer and x2x2 is even, then xx is also even."
To prove its truth via contrapositive, we start by negating both the hypothesis and the conclusion of the original statement:
Original statement: If xx is an integer and x2x2 is even, then xx is also even. Contrapositive: If xx is an integer and xx is not even, then x2x2 is not even.
Now, let's prove the contrapositive statement:
Suppose xx is an integer and xx is not even. This means xx is odd.
If xx is odd, we can express it as x=2k+1x=2k+1, where kk is an integer.
Now, let's find x2x2: x2=(2k+1)2=4k2+4k+1x2=(2k+1)2=4k2+4k+1
This is the expression for an odd number, as it can be represented as 2(2k2+2k)+12(2k2+2k)+1. Thus, x2x2 is odd.
Therefore, the contrapositive statement holds true. Hence, by the method of contrapositive, we have shown that if xx is an integer and x2x2 is even, then xx is also even.
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