How to derive the formula of the sum of this finite series: $sum_{n=M}^N a^n$
$begingroup$
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!
algebra-precalculus summation
edited Dec 10 '18 at 12:16
Brahadeesh
6,51442364
asked Dec 6 '18 at 15:59
KnowledgeKnowledge
113
$endgroup$
add a comment |
$begingroup$
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!
algebra-precalculus summation
edited Dec 10 '18 at 12:16
Brahadeesh
6,51442364
asked Dec 6 '18 at 15:59
KnowledgeKnowledge
113
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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!
algebra-precalculus summation
edited Dec 10 '18 at 12:16
Brahadeesh
6,51442364
asked Dec 6 '18 at 15:59
KnowledgeKnowledge
113
$endgroup$
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
algebra-precalculus summation
edited Dec 10 '18 at 12:16
Brahadeesh
6,51442364
asked Dec 6 '18 at 15:59
KnowledgeKnowledge
113
edited Dec 10 '18 at 12:16
Brahadeesh
6,51442364
asked Dec 6 '18 at 15:59
KnowledgeKnowledge
113
edited Dec 10 '18 at 12:16
Brahadeesh
6,51442364
edited Dec 10 '18 at 12:16
Brahadeesh
6,51442364
edited Dec 10 '18 at 12:16
Brahadeesh
6,51442364
6,51442364
asked Dec 6 '18 at 15:59
KnowledgeKnowledge
113
asked Dec 6 '18 at 15:59
KnowledgeKnowledge
113
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
add a comment |
$begingroup$
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
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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
$$sum_{n=M}^N 1 $$
answered Dec 6 '18 at 16:04
gimusigimusi
93.1k84594
$endgroup$
$begingroup$
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
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3028674%2fhow-to-derive-the-formula-of-the-sum-of-this-finite-series-sum-n-mn-an%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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
$$sum_{n=M}^N 1 $$
answered Dec 6 '18 at 16:04
gimusigimusi
93.1k84594
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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
$$sum_{n=M}^N 1 $$
answered Dec 6 '18 at 16:04
gimusigimusi
93.1k84594
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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
$$sum_{n=M}^N 1 $$
answered Dec 6 '18 at 16:04
gimusigimusi
93.1k84594
$endgroup$
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
$$sum_{n=M}^N 1 $$
answered Dec 6 '18 at 16:04
gimusigimusi
93.1k84594
answered Dec 6 '18 at 16:04
gimusigimusi
93.1k84594
answered Dec 6 '18 at 16:04
gimusigimusi
93.1k84594
answered Dec 6 '18 at 16:04
gimusigimusi
93.1k84594
93.1k84594
$begingroup$
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
add a comment |
$begingroup$
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
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3028674%2fhow-to-derive-the-formula-of-the-sum-of-this-finite-series-sum-n-mn-an%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
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