Answer
Verified
377.1k+ views
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.
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.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE