Arithmetic Sequence

Number Theory Level pending

Consider the arithmetic sequence: t x = a x + c { t }_{ x }=ax+c .

Let t m = n { t }_{ m }=n and t n = m { t }_{ n }=m where n ≠ m . n\neq m.

In simplest form, what is the value of t m + n { t }_{ m+n } ?

0 0 a n m + 1 anm + 1 2 2 a c + n m ac+nm m + n m+n 2 a n m + 2 2anm + 2 2 m n a 2 + 2 a c + c 2mn{ a }^{ 2 }+2ac+c 1 1

Vladimir Smith by Vladimir Smith , 20, Australia

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1 solution

Vladimir Smith
Jan 16, 2016

n = t m = a m + c m = t n = a n + c n − m = a m + c − ( a n + c ) n − m = a m − a n a = − 1 o r n = m n={ t }_{ m }=am+c\\ m={ t }_{ n }=an+c\\ n-m=am+c-(an+c)\\ n-m=am-an\\ a=-1\quad or\quad n=m\\

We eliminate n = m n=m as the question specifies that n ≠ m n\neq m . From the first two equations:

n + m = a ( n + m ) + 2 c n+m=a(n+m)+2c

Substituting a = − 1 a=-1

n + m = − ( n + m ) + 2 c 2 n + 2 m = 2 c c = n + m n+m=-(n+m)+2c\\ 2n+2m=2c\\ c=n+m

t m + n = a m + a n + c t m + n = − m − n + m + n t m + n = 0 { t }_{ m+n }=am+an+c\\ { t }_{ m+n }=-m-n+m+n\\ { t }_{ m+n }=0

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