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.