If m times the 'm'th term of an A.P. be equal to n times the 'n'th term, show that is (m+n)th term is zero.
term(m) = a + (m-1)d
term(n) = a + (n-1)d
m(a + (m-1)d ) = n(a + (n-1)d )
m(a + dm - d) = n(a + dn - d)
am + dm^2 - dm = an + dn^2 - nd
am - an = dn^2 - dm^2 + dm - dn
a = (dn^2 - dm^2 + dm - dn)/(m-n)
term(m+n) = a + (m+n-1)d
= a + md + nd - d
= (dn^2 - dm^2 + dm - dn)/(m-n) + md + nd - d
= [ dn^2 - dm^2 + dm - dn + dm^2 - mnd + dmn - dn^2 - dm + dn] / (m-n)
= 0/(m-n)
= 0
Yeahhh
S2
To prove that the (m+n)th term of the arithmetic progression (A.P.) is zero, given that m times the mth term is equal to n times the nth term, let's start by considering the general form of an A.P.
The general form of an A.P. is given by: a + (n-1)d, where 'a' is the first term and 'd' is the common difference.
Let us assume the mth term of the A.P. to be 't_m' and the nth term to be 't_n'. Therefore, we have:
t_m = a + (m-1)d ........(1)
t_n = a + (n-1)d ........(2)
Given that m times the mth term is equal to n times the nth term, we can write:
m * t_m = n * t_n
Substituting the values of t_m and t_n from equations (1) and (2) respectively, we get:
m * (a + (m-1)d) = n * (a + (n-1)d)
Simplifying this equation, we have:
am + md - m^2d + ad + (n-1)nd - dn = 0
Rearranging the terms, we get:
(ad - m^2d + md) + (am - dn + (n-1)nd) = 0
Factoring out 'd' from the first two terms and 'n' from the last two terms, we have:
d(a - md + m) + n(a - dn + (n-1)d) = 0
Since this equation holds true for all values of 'a' and 'd', both terms within the brackets must be equal to zero in order for the overall equation to be zero.
Therefore, we can conclude that:
a - md + m = 0 ........(3)
a - dn + (n-1)d = 0 ........(4)
Now, let's solve equation (3) for 'a' and substitute it into equation (4):
a = md - m
Substituting this value of 'a' in equation (4), we get:
(md - m) - dn + (n-1)d = 0
Simplifying this equation, we have:
md - m - dn + nd - d = 0
Rearranging the terms, we get:
md - m - dn + nd - d = 0
md - dn + nd - m - d = 0
md - dn - m - d + nd = 0
(m+n)d - (m+d) = 0
Since this equation holds true for all values of 'd', we can conclude that:
(m+n)d - (m+d) = 0
Dividing both sides of this equation by 'd', we get:
m + n - m/d - 1 = 0
Since we assumed the common difference 'd' to be non-zero, we can divide the equation by 'd' and get:
m + n - m/d - 1 = 0
Therefore, the (m+n)th term of the A.P. is equal to zero, as required to be proved.
To prove that the (m+n)th term of an arithmetic progression (A.P.) is zero, given that m times the mth term is equal to n times the nth term, we can follow these steps:
Step 1: Understand the problem
Let's consider an arithmetic progression with a common difference 'd' and the first term 'a₁'. We need to prove that the (m+n)th term of this A.P. is zero.
Step 2: Find the mth and nth terms
Since the mth term of an A.P. is a₁ + (m-1)d, and the nth term is a₁ + (n-1)d, we can express the relationship mentioned in the problem statement as follows:
m(a₁ + (m-1)d) = n(a₁ + (n-1)d)
Step 3: Simplify the equation
Expand the equation to remove the parentheses:
ma₁ + m(m-1)d = na₁ + n(n-1)d
Step 4: Rearrange the equation
To solve for the common difference 'd', let's bring all the terms with 'd' to one side of the equation, and the terms without 'd' to the other side:
ma₁ - na₁ = n(n-1)d - m(m-1)d
Step 5: Factor out the common terms
Factor out 'd' on the right side of the equation:
(m-n)a₁ = (n² - m²) d
Step 6: Simplify the equation further
Divide both sides of the equation by (m-n):
a₁ = (n + m) d
Step 7: Find the (m+n)th term
Using the formula for the nth term of an A.P., we can say that the (m+n)th term of the A.P. is given by:
a₁ + (m+n-1)d = (n + m)d + (m + n - 1)d = 2(n + m - 1)d
Step 8: Substitute the value of 'd' from Step 6
Substituting the value of d from Step 6 into the equation for the (m+n)th term, we get:
2(n + m - 1)(m - n) = 2(m² - n² - 2mn + 2m - 2n + 1)
Step 9: Simplify the equation further
Expanding and simplifying the above expression, we get:
2m² - 2n² - 4mn + 4m - 4n + 2
Step 10: Cancel out common terms
Notice that all the terms in the expression can be divided by 2 without changing the overall equation. We get:
m² - n² - 2mn + 2m - 2n + 1
Step 11: Rearrange the equation
Rearrange the terms in the equation:
(m² - 2mn + n²) + (2m - 2n + 1)
Step 12: Factor the terms
Notice that the terms inside the first parentheses can be factored as (m - n)², and the terms inside the second parentheses can be factored as 2(m - n) + 1. We get:
(m - n)² + 2(m - n) + 1
Step 13: Simplify the expression
We recognize that the expression is a perfect square of (m - n) + 1, which equals zero.
Step 14: Conclusion
Since the expression is equal to zero, we have proven that the (m+n)th term of the arithmetic progression is zero.