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Sum of the first 20 terms of an Arithmetic Progression is equal to the sum of the next 10 terms (i.e., from 21st to 30th) of the same Arithmetic Progression.If all the terms in the AP are natural numbers, which of the following can be the 10th term of the AP?
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Sum of the first 20 terms of an Arithmetic Progression is equal to the...
Problem Analysis:
We are given an arithmetic progression (AP) and we need to find the possible values of its 10th term. The sum of the first 20 terms of the AP is equal to the sum of the next 10 terms.
Solution:
Let's assume that the first term of the AP is 'a' and the common difference is 'd'.
Sum of the first 20 terms:
The sum of an AP can be calculated using the formula:
Sn = (n/2) * (2a + (n-1)d)
For the first 20 terms, n = 20. Therefore, the sum of the first 20 terms can be written as:
S20 = (20/2) * (2a + 19d) = 10(2a + 19d)
Sum of the next 10 terms:
For the next 10 terms, we need to consider the terms from 21st to 30th.
Let's calculate the sum of these terms using the same formula as before:
S10 = (10/2) * (2a + (10-1)d) = 5(2a + 9d)
Equation:
According to the problem statement, the sum of the first 20 terms is equal to the sum of the next 10 terms. Therefore, we can write the equation as:
10(2a + 19d) = 5(2a + 9d)
Simplifying the Equation:
Let's simplify the equation to get rid of the brackets and combine like terms:
20a + 190d = 10a + 45d
Subtracting 10a and 45d from both sides of the equation, we get:
10a + 145d = 0
Possible Values of the 10th Term:
Now let's find the possible values of the 10th term by considering different values of 'a' and 'd'.
Case 1: a = 0
If we substitute a = 0 in the equation, we get:
0 + 145d = 0
This implies that d can be any value, as long as it is a natural number. Therefore, the 10th term can be any natural number.
Case 2: a ≠ 0
If we assume a ≠ 0, we can simplify the equation further:
10 + 145d = 0
This equation has no solution because there is no natural number 'd' that satisfies the equation. Therefore, in this case, there are no possible values for the 10th term.
Therefore, the possible values of the 10th term of the AP are all the natural numbers if a = 0, and there are no possible values if a ≠ 0.
We are given an arithmetic progression (AP) and we need to find the possible values of its 10th term. The sum of the first 20 terms of the AP is equal to the sum of the next 10 terms.
Solution:
Let's assume that the first term of the AP is 'a' and the common difference is 'd'.
Sum of the first 20 terms:
The sum of an AP can be calculated using the formula:
Sn = (n/2) * (2a + (n-1)d)
For the first 20 terms, n = 20. Therefore, the sum of the first 20 terms can be written as:
S20 = (20/2) * (2a + 19d) = 10(2a + 19d)
Sum of the next 10 terms:
For the next 10 terms, we need to consider the terms from 21st to 30th.
Let's calculate the sum of these terms using the same formula as before:
S10 = (10/2) * (2a + (10-1)d) = 5(2a + 9d)
Equation:
According to the problem statement, the sum of the first 20 terms is equal to the sum of the next 10 terms. Therefore, we can write the equation as:
10(2a + 19d) = 5(2a + 9d)
Simplifying the Equation:
Let's simplify the equation to get rid of the brackets and combine like terms:
20a + 190d = 10a + 45d
Subtracting 10a and 45d from both sides of the equation, we get:
10a + 145d = 0
Possible Values of the 10th Term:
Now let's find the possible values of the 10th term by considering different values of 'a' and 'd'.
Case 1: a = 0
If we substitute a = 0 in the equation, we get:
0 + 145d = 0
This implies that d can be any value, as long as it is a natural number. Therefore, the 10th term can be any natural number.
Case 2: a ≠ 0
If we assume a ≠ 0, we can simplify the equation further:
10 + 145d = 0
This equation has no solution because there is no natural number 'd' that satisfies the equation. Therefore, in this case, there are no possible values for the 10th term.
Therefore, the possible values of the 10th term of the AP are all the natural numbers if a = 0, and there are no possible values if a ≠ 0.
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Sum of the first 20 terms of an Arithmetic Progression is equal to the sum of the next 10 terms (i.e., from 21st to 30th) of the same Arithmetic Progression.If all the terms in the AP are natural numbers, which of the following can be the 10th term of the AP?
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Sum of the first 20 terms of an Arithmetic Progression is equal to the sum of the next 10 terms (i.e., from 21st to 30th) of the same Arithmetic Progression.If all the terms in the AP are natural numbers, which of the following can be the 10th term of the AP? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Sum of the first 20 terms of an Arithmetic Progression is equal to the sum of the next 10 terms (i.e., from 21st to 30th) of the same Arithmetic Progression.If all the terms in the AP are natural numbers, which of the following can be the 10th term of the AP? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Sum of the first 20 terms of an Arithmetic Progression is equal to the sum of the next 10 terms (i.e., from 21st to 30th) of the same Arithmetic Progression.If all the terms in the AP are natural numbers, which of the following can be the 10th term of the AP?.
Sum of the first 20 terms of an Arithmetic Progression is equal to the sum of the next 10 terms (i.e., from 21st to 30th) of the same Arithmetic Progression.If all the terms in the AP are natural numbers, which of the following can be the 10th term of the AP? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Sum of the first 20 terms of an Arithmetic Progression is equal to the sum of the next 10 terms (i.e., from 21st to 30th) of the same Arithmetic Progression.If all the terms in the AP are natural numbers, which of the following can be the 10th term of the AP? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Sum of the first 20 terms of an Arithmetic Progression is equal to the sum of the next 10 terms (i.e., from 21st to 30th) of the same Arithmetic Progression.If all the terms in the AP are natural numbers, which of the following can be the 10th term of the AP?.
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