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If the nth term of AP is m and the nth term of the same AP is m, then (m + n)th term of AP is
- a)m+n
- b)0
- c)m-n
- d)-m+n
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
If the nth term of AP is m and the nth term of the same AP is m, then ...
Understanding the Arithmetic Progression (AP)
In an arithmetic progression (AP), the nth term can be expressed as:
Tn = a + (n - 1)d
where 'a' is the first term, 'd' is the common difference, and 'n' is the term number.
Given Information
- The nth term of the AP is m:
Tn = m ⇒ a + (n - 1)d = m.
- The mth term of the same AP is also m:
Tm = m ⇒ a + (m - 1)d = m.
Setting Up the Equations
From the two equations:
1. a + (n - 1)d = m
2. a + (m - 1)d = m
By subtracting the first equation from the second:
(a + (m - 1)d) - (a + (n - 1)d) = 0
This simplifies to:
(m - 1)d - (n - 1)d = 0
which gives us:
(m - n)d = 0.
This means either m = n (same term) or d = 0 (constant sequence).
Finding the (m + n)th Term
To find the (m + n)th term:
T(m+n) = a + (m + n - 1)d.
By substituting 'a' from either of the previous equations:
Let’s take a from Tn:
a = m - (n - 1)d.
Now substituting into T(m+n):
T(m+n) = [m - (n - 1)d] + (m + n - 1)d
= m + (m + n - 1)d - (n - 1)d
= m + (m + n - 1 - n + 1)d
= m + md
= m + 0 (if d=0)
= 0.
Thus, the (m + n)th term of the AP is 0.
Conclusion
The correct answer is option 'B': 0.
In an arithmetic progression (AP), the nth term can be expressed as:
Tn = a + (n - 1)d
where 'a' is the first term, 'd' is the common difference, and 'n' is the term number.
Given Information
- The nth term of the AP is m:
Tn = m ⇒ a + (n - 1)d = m.
- The mth term of the same AP is also m:
Tm = m ⇒ a + (m - 1)d = m.
Setting Up the Equations
From the two equations:
1. a + (n - 1)d = m
2. a + (m - 1)d = m
By subtracting the first equation from the second:
(a + (m - 1)d) - (a + (n - 1)d) = 0
This simplifies to:
(m - 1)d - (n - 1)d = 0
which gives us:
(m - n)d = 0.
This means either m = n (same term) or d = 0 (constant sequence).
Finding the (m + n)th Term
To find the (m + n)th term:
T(m+n) = a + (m + n - 1)d.
By substituting 'a' from either of the previous equations:
Let’s take a from Tn:
a = m - (n - 1)d.
Now substituting into T(m+n):
T(m+n) = [m - (n - 1)d] + (m + n - 1)d
= m + (m + n - 1)d - (n - 1)d
= m + (m + n - 1 - n + 1)d
= m + md
= m + 0 (if d=0)
= 0.
Thus, the (m + n)th term of the AP is 0.
Conclusion
The correct answer is option 'B': 0.
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Community Answer
If the nth term of AP is m and the nth term of the same AP is m, then ...
- Tn = m = a + (n - l ) d
- Tm = n= a + (m - l ) d
- subtracting them we get d=-1
- and a=m+n-1
- Add the two and solve through the options given.
- then (m + n)th term of AP= a+(m+n-1)d
- Putting values of a and d in the solution, we get
- = m+n-1+(m+n-1)x-1
- = 0
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