How to derive the formula of the sum of this finite series: $sum_{n=M}^N a^n$












1












$begingroup$


I would like to know how to arrive at the following result that
my teacher wrote on the board. They did not explain how it was done. I am also not sure what this series is called. Is it perhaps a power series?




$$N>M :sum_{n=M}^N a^n = frac{a^M-a^1a^n}{1-a},aneq1$$
$$N>M: sum_{n=M}^N a^n = N-M+1,a=1$$




I am quite lost since my teacher only wrote the above formulae without any derivation. Can someone help me understand why they are true? Thank you!










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  • $begingroup$
    Looks like an application of geometric series, i.e. $sum_{i=0}^n a^i = frac{1-a^{n+1}}{1-a}$.
    $endgroup$
    – TrostAft
    Dec 6 '18 at 16:01


















1












$begingroup$


I would like to know how to arrive at the following result that
my teacher wrote on the board. They did not explain how it was done. I am also not sure what this series is called. Is it perhaps a power series?




$$N>M :sum_{n=M}^N a^n = frac{a^M-a^1a^n}{1-a},aneq1$$
$$N>M: sum_{n=M}^N a^n = N-M+1,a=1$$




I am quite lost since my teacher only wrote the above formulae without any derivation. Can someone help me understand why they are true? Thank you!










share|cite|improve this question











$endgroup$












  • $begingroup$
    Looks like an application of geometric series, i.e. $sum_{i=0}^n a^i = frac{1-a^{n+1}}{1-a}$.
    $endgroup$
    – TrostAft
    Dec 6 '18 at 16:01
















1












1








1





$begingroup$


I would like to know how to arrive at the following result that
my teacher wrote on the board. They did not explain how it was done. I am also not sure what this series is called. Is it perhaps a power series?




$$N>M :sum_{n=M}^N a^n = frac{a^M-a^1a^n}{1-a},aneq1$$
$$N>M: sum_{n=M}^N a^n = N-M+1,a=1$$




I am quite lost since my teacher only wrote the above formulae without any derivation. Can someone help me understand why they are true? Thank you!










share|cite|improve this question











$endgroup$




I would like to know how to arrive at the following result that
my teacher wrote on the board. They did not explain how it was done. I am also not sure what this series is called. Is it perhaps a power series?




$$N>M :sum_{n=M}^N a^n = frac{a^M-a^1a^n}{1-a},aneq1$$
$$N>M: sum_{n=M}^N a^n = N-M+1,a=1$$




I am quite lost since my teacher only wrote the above formulae without any derivation. Can someone help me understand why they are true? Thank you!







algebra-precalculus summation






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edited Dec 10 '18 at 12:16









Brahadeesh

6,32942363




6,32942363










asked Dec 6 '18 at 15:59









KnowledgeKnowledge

113




113












  • $begingroup$
    Looks like an application of geometric series, i.e. $sum_{i=0}^n a^i = frac{1-a^{n+1}}{1-a}$.
    $endgroup$
    – TrostAft
    Dec 6 '18 at 16:01




















  • $begingroup$
    Looks like an application of geometric series, i.e. $sum_{i=0}^n a^i = frac{1-a^{n+1}}{1-a}$.
    $endgroup$
    – TrostAft
    Dec 6 '18 at 16:01


















$begingroup$
Looks like an application of geometric series, i.e. $sum_{i=0}^n a^i = frac{1-a^{n+1}}{1-a}$.
$endgroup$
– TrostAft
Dec 6 '18 at 16:01






$begingroup$
Looks like an application of geometric series, i.e. $sum_{i=0}^n a^i = frac{1-a^{n+1}}{1-a}$.
$endgroup$
– TrostAft
Dec 6 '18 at 16:01












1 Answer
1






active

oldest

votes


















1












$begingroup$

The first one can be derived by the geometric series



$$sum_{n=M}^N a^n =sum_{n=0}^N a^n-sum_{n=0}^{M-1} a^n $$



the second one is simply



$$sum_{n=M}^N 1 $$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    but you wrote from 0 to N i dont think its the same, sum of series for exm. from 5 till 8 its not like from 0 to 5 minus series from 0 to 7.
    $endgroup$
    – Knowledge
    Dec 6 '18 at 16:12










  • $begingroup$
    @DvirIhie But form $5$ to $8$ is like $0$ to $8$ minus $0$ to $4$, that's what I wrote.
    $endgroup$
    – gimusi
    Dec 6 '18 at 16:16










  • $begingroup$
    oh my bad thank you !!!
    $endgroup$
    – Knowledge
    Dec 6 '18 at 16:18










  • $begingroup$
    @DvirIhie You are welcome! Bye
    $endgroup$
    – gimusi
    Dec 6 '18 at 16:19











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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

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active

oldest

votes









1












$begingroup$

The first one can be derived by the geometric series



$$sum_{n=M}^N a^n =sum_{n=0}^N a^n-sum_{n=0}^{M-1} a^n $$



the second one is simply



$$sum_{n=M}^N 1 $$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    but you wrote from 0 to N i dont think its the same, sum of series for exm. from 5 till 8 its not like from 0 to 5 minus series from 0 to 7.
    $endgroup$
    – Knowledge
    Dec 6 '18 at 16:12










  • $begingroup$
    @DvirIhie But form $5$ to $8$ is like $0$ to $8$ minus $0$ to $4$, that's what I wrote.
    $endgroup$
    – gimusi
    Dec 6 '18 at 16:16










  • $begingroup$
    oh my bad thank you !!!
    $endgroup$
    – Knowledge
    Dec 6 '18 at 16:18










  • $begingroup$
    @DvirIhie You are welcome! Bye
    $endgroup$
    – gimusi
    Dec 6 '18 at 16:19
















1












$begingroup$

The first one can be derived by the geometric series



$$sum_{n=M}^N a^n =sum_{n=0}^N a^n-sum_{n=0}^{M-1} a^n $$



the second one is simply



$$sum_{n=M}^N 1 $$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    but you wrote from 0 to N i dont think its the same, sum of series for exm. from 5 till 8 its not like from 0 to 5 minus series from 0 to 7.
    $endgroup$
    – Knowledge
    Dec 6 '18 at 16:12










  • $begingroup$
    @DvirIhie But form $5$ to $8$ is like $0$ to $8$ minus $0$ to $4$, that's what I wrote.
    $endgroup$
    – gimusi
    Dec 6 '18 at 16:16










  • $begingroup$
    oh my bad thank you !!!
    $endgroup$
    – Knowledge
    Dec 6 '18 at 16:18










  • $begingroup$
    @DvirIhie You are welcome! Bye
    $endgroup$
    – gimusi
    Dec 6 '18 at 16:19














1












1








1





$begingroup$

The first one can be derived by the geometric series



$$sum_{n=M}^N a^n =sum_{n=0}^N a^n-sum_{n=0}^{M-1} a^n $$



the second one is simply



$$sum_{n=M}^N 1 $$






share|cite|improve this answer









$endgroup$



The first one can be derived by the geometric series



$$sum_{n=M}^N a^n =sum_{n=0}^N a^n-sum_{n=0}^{M-1} a^n $$



the second one is simply



$$sum_{n=M}^N 1 $$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 6 '18 at 16:04









gimusigimusi

92.8k84494




92.8k84494












  • $begingroup$
    but you wrote from 0 to N i dont think its the same, sum of series for exm. from 5 till 8 its not like from 0 to 5 minus series from 0 to 7.
    $endgroup$
    – Knowledge
    Dec 6 '18 at 16:12










  • $begingroup$
    @DvirIhie But form $5$ to $8$ is like $0$ to $8$ minus $0$ to $4$, that's what I wrote.
    $endgroup$
    – gimusi
    Dec 6 '18 at 16:16










  • $begingroup$
    oh my bad thank you !!!
    $endgroup$
    – Knowledge
    Dec 6 '18 at 16:18










  • $begingroup$
    @DvirIhie You are welcome! Bye
    $endgroup$
    – gimusi
    Dec 6 '18 at 16:19


















  • $begingroup$
    but you wrote from 0 to N i dont think its the same, sum of series for exm. from 5 till 8 its not like from 0 to 5 minus series from 0 to 7.
    $endgroup$
    – Knowledge
    Dec 6 '18 at 16:12










  • $begingroup$
    @DvirIhie But form $5$ to $8$ is like $0$ to $8$ minus $0$ to $4$, that's what I wrote.
    $endgroup$
    – gimusi
    Dec 6 '18 at 16:16










  • $begingroup$
    oh my bad thank you !!!
    $endgroup$
    – Knowledge
    Dec 6 '18 at 16:18










  • $begingroup$
    @DvirIhie You are welcome! Bye
    $endgroup$
    – gimusi
    Dec 6 '18 at 16:19
















$begingroup$
but you wrote from 0 to N i dont think its the same, sum of series for exm. from 5 till 8 its not like from 0 to 5 minus series from 0 to 7.
$endgroup$
– Knowledge
Dec 6 '18 at 16:12




$begingroup$
but you wrote from 0 to N i dont think its the same, sum of series for exm. from 5 till 8 its not like from 0 to 5 minus series from 0 to 7.
$endgroup$
– Knowledge
Dec 6 '18 at 16:12












$begingroup$
@DvirIhie But form $5$ to $8$ is like $0$ to $8$ minus $0$ to $4$, that's what I wrote.
$endgroup$
– gimusi
Dec 6 '18 at 16:16




$begingroup$
@DvirIhie But form $5$ to $8$ is like $0$ to $8$ minus $0$ to $4$, that's what I wrote.
$endgroup$
– gimusi
Dec 6 '18 at 16:16












$begingroup$
oh my bad thank you !!!
$endgroup$
– Knowledge
Dec 6 '18 at 16:18




$begingroup$
oh my bad thank you !!!
$endgroup$
– Knowledge
Dec 6 '18 at 16:18












$begingroup$
@DvirIhie You are welcome! Bye
$endgroup$
– gimusi
Dec 6 '18 at 16:19




$begingroup$
@DvirIhie You are welcome! Bye
$endgroup$
– gimusi
Dec 6 '18 at 16:19


















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