Question

If the sum of first m terms of an AP is the same as the sum of its first n terms, show

that the sum of its first (m+n) terms is zero​

Answer 1

S(m+n)=0

hence proved

Step-by-step explanation:

Sm=Sn

m/2 (2a +(m-1)d)=n/2 (2a +(n-1)d) =0

in both side 2 is cancelled

a(m-n)+ d [ (m^2 -m) + (-n^2+n)] =0

a(m-n) +d [ ( m^2- n^2)- (m-n)] =0

a(m-n) +d [( m-n)(m+n) - (m-n)=0

(m-n) [ a + d( m+n-1) ] =0

(m-n) will go on right hand side and become 0

now value of a = - d(m+n-1)__________(1)

now using value of 1 in S(m+n)

(m+n)/2 (a+(m +n-1)d )=0__________(2)

from 1 into 2

(m+n)/2 ( -d(m+n-1) +d(m+n-1) ) = 0

(m+n)/2 (0) =0

0=0

LHS=RHS

(hence proved)

Answer 2

$\huge{ \underline{ \overline{ \mid{ \mathfrak{ \red{ \: \: SOLUTION \: \: } \mid}}}}}$

$\underline{ \mathfrak{ \: \: Given : \: }} \\ \\ \: \: \text{Sum of first m terms of an A.P. is same as} \\ \text{the sum \: of first n terms, where m \: is \: not \: } \\ \text{equal \:to n }$

$\underline{ \mathfrak{ \: \: To \: \: show : \mapsto \: }} \\ \\ \text{Sum of first (m+n) terms is equal to zero. }$

$\mathfrak{ \: Suppose,} \\ \\ \: \: \: \: \: \: \: \: \: \text{First term of the A.P. = a} \\ \\ \: \: \: \: \: \: \: \: \: \text{Common difference = d} \\ \\ \: \: \: \: \: \: \: \: \tt{Sum \: \: of \: \: first \: \: m \: \: terms = S_m} \\ \text{And,} \\ \: \: \: \: \: \: \: \: \tt{Sum \: \: of \: \: first \: \: n \: \: terms = S_n}$

$\underline{ \bold{ \: \: A.T.Q., \: \: }} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\: \: \: \: \: \huge{ \tt{S_m = S_n } }\\ \\ \tt{ \implies \: \frac{m}{ \cancel{2}} \{2a + (m - 1)d \} = \frac{n}{ \cancel{2}} \{2a + (n - 1)d \} } \\ \\ \tt{ \implies \: m \{2a + (m - 1)d \} = n \{2a + (n - 1)d \}} \\ \\ \tt{ \implies \: m \times 2a + m \times (m - 1)d = n\times 2a + n \times (n - 1)d} \\ \\ \tt{ \implies \: m \times 2a - n \times 2a = \{n \times (n - 1)d \} - \{m \times (m- 1)d \}} \\ \\ \tt{ \implies \:2a \times (m - n) = \{n \times (nd - d) \} - \{m \times (md - d) \}} \\ \\ \tt{ \implies \:2a \times ( m- n) = (n {}^{2}d - nd) - (m{}^{2}d - md) } \\ \\ \tt{ \implies \:2a \times ( m- n) = n {}^{2}d - nd - m{}^{2}d + md } \\ \\ \tt{ \implies \: 2a \times (m - n) = n {}^{2}d - m{}^{2}d - nd +md } \\ \\ \tt{ \implies \: 2a \times (m - n) = d(n {}^{2} - m {}^{2} ) - d(n - m)} \\ \\ \tt{ \implies \: 2a \times (m - n) = d \{( n+ m)(n - m) \} - d(n - m)} \\ \\ \tt{ \implies \: 2a \times (m - n) = d(n - m) \{(n + m) - 1 \} } \\ \\ \tt{ \implies \: 2a \times \cancel{(m - n) }= - \: d \: \cancel{(m - n)} \{(m + n) - 1 \}} \\ \\ \tt{ \implies \: 2a = - d \{(m + n) - 1 \}} \\ \\ \tt{ \implies \: 2a + d \{(m + n) - 1 \} = 0} \\ \\ \tt{ \implies \: 2a + \{(m + n) - 1 \} d= 0 \: \: \: - - - - - > (1)}$

$\mathfrak{ \:Now, \: } \\ \\ \underline{ \mathfrak{ \: Suppose, \: }} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \tt{The \: \: sum \: \: of \: \: first \: \: (m+n) \: \: terms = S_{(m+n)}}$

$\tt{ \therefore{ \: S_{(m+n)} = \frac{( m+ n)}{2}[ 2a + \{(m + n) - 1 \}d}]} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt{ = \frac{(m + n}{2} \times 0 \: \: \: \: \: \: \: \: \:[From \: \: (1)] } \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt{ =0}$

$\: \: \: \: \: \: \: \: \: \tt{ \therefore \: \: S _{(m + n)} = 0 \: \: \: \: \red{ i.e.} \: \: Sum \: \: of \: \: first \: \: } \\ \\ \tt{ (m+n) terms \: \: is \: \: equal \: \: to \: \: zero. } \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \red{\underline{ \text{ \: \: Showed.\: \: }}}$