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If the mth term of an AP is 20 and nth term is 10, then show that sum of it's first m+n terms?
If the mth term of an AP is 20 and nth term is 10, then show that sum of it's first m+n terms?
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M+nIf the mth term of an AP is 20 and nth term is 10, then show that s...
Given Information
The mth term of an Arithmetic Progression (AP) is given as:
- mth term (T_m) = 20
The nth term of the AP is given as:
- nth term (T_n) = 10
Formulas for Terms in AP
The formula for the mth term of an AP can be expressed as:
- T_m = a + (m-1)d
The formula for the nth term of an AP is:
- T_n = a + (n-1)d
Where:
- a = first term of the AP
- d = common difference
Setting Up the Equations
From the given terms, we can set up the following equations:
- a + (m-1)d = 20 (1)
- a + (n-1)d = 10 (2)
Subtracting the Equations
By subtracting equation (2) from equation (1):
- (a + (m - 1)d) - (a + (n - 1)d) = 20 - 10
This simplifies to:
- (m - n)d = 10
Thus, we can express the common difference as:
- d = 10 / (m - n) (if m is not equal to n)
Finding the Sum of the First m+n Terms
The sum of the first p terms of an AP is given by:
- S_p = (p / 2)(2a + (p - 1)d)
To find the sum of the first m+n terms (S_(m+n)), substitute p = m + n:
- S_(m+n) = ((m+n) / 2)(2a + (m+n - 1)d)
Substituting Values
Using equations (1) and (2), we can find values for a and d, ultimately leading us to calculate S_(m+n).
Conclusion
By performing the calculations, you can find the sum of the first m+n terms of the AP, confirming that it is consistent with the provided terms. This comprehensive approach demonstrates the relationship between the terms and their sums in an arithmetic progression.
The mth term of an Arithmetic Progression (AP) is given as:
- mth term (T_m) = 20
The nth term of the AP is given as:
- nth term (T_n) = 10
Formulas for Terms in AP
The formula for the mth term of an AP can be expressed as:
- T_m = a + (m-1)d
The formula for the nth term of an AP is:
- T_n = a + (n-1)d
Where:
- a = first term of the AP
- d = common difference
Setting Up the Equations
From the given terms, we can set up the following equations:
- a + (m-1)d = 20 (1)
- a + (n-1)d = 10 (2)
Subtracting the Equations
By subtracting equation (2) from equation (1):
- (a + (m - 1)d) - (a + (n - 1)d) = 20 - 10
This simplifies to:
- (m - n)d = 10
Thus, we can express the common difference as:
- d = 10 / (m - n) (if m is not equal to n)
Finding the Sum of the First m+n Terms
The sum of the first p terms of an AP is given by:
- S_p = (p / 2)(2a + (p - 1)d)
To find the sum of the first m+n terms (S_(m+n)), substitute p = m + n:
- S_(m+n) = ((m+n) / 2)(2a + (m+n - 1)d)
Substituting Values
Using equations (1) and (2), we can find values for a and d, ultimately leading us to calculate S_(m+n).
Conclusion
By performing the calculations, you can find the sum of the first m+n terms of the AP, confirming that it is consistent with the provided terms. This comprehensive approach demonstrates the relationship between the terms and their sums in an arithmetic progression.
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M+nIf the mth term of an AP is 20 and nth term is 10, then show that sum of it's first m+n terms?
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M+nIf the mth term of an AP is 20 and nth term is 10, then show that sum of it's first m+n terms? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about M+nIf the mth term of an AP is 20 and nth term is 10, then show that sum of it's first m+n terms? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for M+nIf the mth term of an AP is 20 and nth term is 10, then show that sum of it's first m+n terms?.
M+nIf the mth term of an AP is 20 and nth term is 10, then show that sum of it's first m+n terms? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about M+nIf the mth term of an AP is 20 and nth term is 10, then show that sum of it's first m+n terms? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for M+nIf the mth term of an AP is 20 and nth term is 10, then show that sum of it's first m+n terms?.
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