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In an A.P., $ {m^{th}} $ term is n and $ {n^{th}} $ term is m, show that its $ {r^{th}} $ term is $ (m + n - r). $

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Last updated date: 13th Jul 2024
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Answer
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Hint: An Arithmetic Progression (AP) is the sequence of numbers in which the difference of two successive numbers is always constant.
The standard formula for Arithmetic Progression is: $ {a_n} = a + (n - 1)d $
Where $ {t_n} = $ nth term in the AP
 $ a = $ First term of AP
 $ d = $ Common difference in the series
 $ n = $ Number of terms in the AP
Here, will place $ {m^{th}} $ term is equal to n and $ {n^{th}} $ term is equal to m

Complete step-by-step answer:
Now, take the standard formula and find $ {m^{th}} $ term and $ {n^{th}} $ term
 $ {a_m} = a + (m - 1)d $
Given that - $ {a_m} = n $
Place in the above equation –
 $ n = a + (m - 1)d $ …… (A)
Similarly, $ {a_n} = a + (n - 1)d $
Given that - $ {a_n} = m $
Place in the above equation –
 $ m = a + (n - 1)d $ …… (B)
Subtract equation (B) from the equation (A)
 $ n - m = a + (m - 1)d - [a + (n - 1)d] $
When there is a negative sign outside the bracket then the sign of the terms inside the brackets changes. Positive term changes to negative and vice-versa.
 $ n - m = a + md - d - a - nd + d $
Combine the like terms in the above expression –
 $ n - m = a - a - d + d + md - nd $
Like terms with the same value and the opposite sign cancels each other.
 $ n - m = md - nd $
Find the common multiple –
 $ n - m = - d(n - m) $
Term multiplicative at one side if moved to the opposite side then it goes in the denominator.
 $ \dfrac{{n - m}}{{ - (n - m)}} = d $
Common factors from the numerator and the denominator cancel each other.
 $ d = ( - 1) $ …. (C)
Place the above value in the equation (A)
 $ n = a + (m - 1)( - 1) $
Simplify the above expression –
 $ n = a - m + 1 $
 $ a = n + m - 1 $ …. (D)
Now, the rth term of the Arithmetic progression is given by –
 $ {a_r} = a + (r - 1)d $
Place the values of equation (C) and (D) in the above equation –
 $ {a_r} = n + m - 1 + (r - 1)( - 1) $
Simplify the above expression –
 $
  {a_r} = n + m - 1 + ( - r + 1) \\
  {a_r} = n + m - 1 - r + 1 \;
  $
Like terms with the same value and opposite sign cancels each other.
 $ {a_r} = m + n - r $
This is the required solution.

Note: Be careful about the sign convention when there is positive term outside the bracket then the sign of the terms inside the bracket remains the same while if there is negative sign outside the bracket then the sign of the terms inside the bracket changes. Positive term changes to negative and negative changes to positive.