CBSE Class 11 > Class 11 Videos > Sum of terms of AP Arithematic Progression - Sequences & Series
Sum of terms of AP Arithematic Progression - Sequences & Series Video Lecture
FAQs on Sum of terms of AP Arithematic Progression - Sequences & Series
| 1. What is an arithmetic progression (AP)? |  |
Ans. An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference (d).
| 2. How do you find the sum of terms in an arithmetic progression (AP)? | |
Ans. To find the sum of terms in an arithmetic progression (AP), we use the formula Sn = (n/2)(2a + (n-1)d), where Sn is the sum of the first n terms, a is the first term, n is the number of terms, and d is the common difference.
| 3. What is the formula for the nth term of an arithmetic progression (AP)? | |
Ans. The formula for the nth term of an arithmetic progression (AP) is given by the formula An = a + (n-1)d, where An is the nth term, a is the first term, n is the term number, and d is the common difference.
| 4. How can we identify if a sequence of numbers is an arithmetic progression (AP)? | |
Ans. To identify if a sequence of numbers is an arithmetic progression (AP), we need to check if the difference between any two consecutive terms is constant. If the difference is constant, then the sequence is an arithmetic progression.
| 5. Can the common difference (d) in an arithmetic progression (AP) be negative? | |
Ans. Yes, the common difference (d) in an arithmetic progression (AP) can be negative. The common difference represents the change between consecutive terms, and this change can be positive or negative.
Text Transcript from Video
hello friends this video sequence and
sale is part 5 is brought you by exam
fear calm no more fear from exam before
watching this video please make sure
that you have watched video part 1 -
part 4 let me introduce some optimally
in real life we have scenarios where you
need to find some up comes maybe example
let's suppose your teacher asked to
increase the number of questions you
solve per day by one who is ample today
you have solved one questions to morazán
proportions then three pushers and four
portions you keep increasing the calm
daily you increase by one and then you
want to find the number of questions you
have solved in the past hundred days so
it's different if your'e give you keep
adding one number one by one it is very
difficult to find answer because we have
got the formulas to find some of such
terms with easily the sum of terms Navy
is represented by n by 2 to a plus n
minus 1 D so let me tell you what are
the values n is the number of terms by
xanthine in this case you want to find
that 107 100 sum of 100 companies the
number of pushes we have solved in
hundred days then n will be 100 is the
first term so in this case would be 1
and D is the common difference in this
case it will also be 1 because every day
you are asked to increase your convent 1
also some of the terms in it we can also
become write n by 2 a plus L where L is
the last term's is I am well one the 579
and you know that this is till 179
this is a sequence of all odd-numbered
you know this value and you can find in
order to find it easier so you can them
say this is V and then by group do you
listen you similar to this because when
I'm saying n by 2 to M plus n minus 1 DS
I can
in this fashion and buy food cave less a
plus in Vegas 1d a plus n minus one D is
nothing but an extra what is n by 2 a
plus B and this is nothing but the small
same family I tried the mail the most
crucial formula is this so please
remember his formula some abdominal b is
nothing but n by Q into a plus n minus 1
D can he grooms formula for a week but
please commence nom nom some of them in
F is nothing but n by 2 into 2 M plus n
minus 1 D let's take for example we have
to find the sum of terms where a 3 6 9
12 15 za v first let's confirm where
this VPN on 3 6 minus 3 is 3 9 minus 6
is 3 12 minus 9 is 3 K minus 2 n is 3 so
it control that's a ap now I'm going to
find the number of terms 1 2 3 4 5 so
that means n is equal to 5
the first term is 3 is equal to see the
difference is 3 so D is the C so we have
M a and D we know that sum is nothing
but n by P into a Lance n minus 1 into D
so we can just fill in the values 5 by 2
into 2 into 3 plus 5 minus 1 is equal to
4 into D is equal to 3 so what you get
is 5 by 2 into 6 plus 8 6 plus 8 is 6
plus 12 this is 6 plus 12 is equal to e
teams are writing it in here and this
can sing this will 9 so 9 to 5 is going
to be 5 so sum of terms of this is 45
you can do a cross verification also
3+6 9 9 plus 9 18 18 plus 12 is 30 30
plus 15 is working 45 so when we do my
evaluation and we do the calculation
using formula what gives
that means our farmlands correct please
note some of the terms is n by 2 into
wave less n minus 1 D toggle proof
scholar using induction so formula says
sum of n term is equal to n by 2 to a
plus n minus 1 of D is the formula where
the sequence is of this form e e + D E +
- d e + 3 d continue is the sequence
what is the CDs consequence because
sequence is nothing but sum of series so
this sequence series nothing but sum of
sequence so this is sequence a a + D if
the story hitler's clearly now s/m is
going to invite to it we in the second -
1 D we have to prove by induction so to
prove by nation' what we do is we need
we as you we prove that this statement
is true for n is equal to 1 if it is
true for n squared - 1 we assume that
this statement is true for n is equal to
K and then we prove that statement is
true for n is equal to ki plus 1 so if
you can do these 3 steps that means
they're formalized consider to be ruled
by nature and is equal to 1 is true and
this vector K as you this it is true and
then we prove that it is true for air is
let you keep this one similarly we do
here s 1 is where n is equal to 1 we do
1 by 2 into into a plus 1 minus 1 is 0
into D so this becomes 1 by 2 into 2 8
and that becomes shape
so we see there our first immensity so
that means this is 2 now we'll assume
that s of K is 2 that is we will do that
as of K is equal to K by 2 into 2 a plus
K minus 1 into D then assume that this
is true now s of T plus 1
finger s of K plus T of n plus 1 or k
plus M because till K you have added
this plus reading T of T plus 1 all
right so this becomes T by 2 into 2 n
plus K minus 1 into D plus T of T plus 1
is nothing but a plus K plus 1 minus 1
into D plus 1 minus 1 castle that is a
plus K so if you solve this this becomes
K by 2 into 2 a this becomes ki a plus K
by 2 into K T this becomes K squared e
minus K by 2 into minus D this becomes K
2 D and this is plus a plus KD all right
so what we get here is what we get here
is he a plus K squared D by 2 this and
this we can add them scale D by 2 plus
KD by 2 plus a or we'll say we take a
common hand we'll put all the things you
see here this K a plus a plus we can say
K Square D by 2 plus KD by 2 so what we
get here is we take a common this
becomes T plus 1 into k plus 1 plus we
take here ki he D by 2 common into k
plus 1 this becomes nothing but he plus
1 into a plus KD by 2 this is the value
we got and this is nothing but s of K
plus one you can show you s of ki plus 1
using the formula it will become K by 2
k plus 1 by 2 into 2 e plus k plus 1
minus 1 into D that is gone this becomes
K plus 1 by 2 into so this becomes K
plus 1 or I can say k plus 1 into and
take to power here plus k dy so if you
see this and this what we have got our
same thus Y you can prove that this
formula is true using induction so
induction with induction we can certify
that this formula is true what we have
done we have proved that this is true
for n is equal to 1 if assume that this
is true for n is equal to K here and
then we have clue that this is true for
n is equal to k plus 1 thus we have to
the salvation formula that is SN is
equal to n by 2 into V Plus M minus 1 by
n minus 1 into D using that chunk now
let's try to prove the same thing using
logic so we'll write the sequence as you
can write the sequence as a this becomes
a plus D this becomes a plus 2 D a + 3 d
a plus holding dot dot dot a plus n
minus 2 d loves a plus n minus 1 because
SN is the last time is a plus n minus 1
T this is the sequence eviler also the
same sequence we can write like this in
the reverse order so let us write the
same sequence in the reverse order so
first we'll write a plus n minus 1 D
this will go me plus n minus 2 D correct
this here will write
plus n minus three here light a plus n
minus 40 this will become a plus n minus
5 D and this will become e this will
become eat less the same sequence I have
done in the reverse direction so you be
add these two you add these two this
becomes 2 SN this is equal to e plus a
plus n minus 1 D but you get a plus 2 a
plus n minus 1 B similarly e Plus D plus
a plus n minus 2 D this also becomes a
plus h 2 e M minus 2 plus 1 become n
minus 1 D similarly here also if you add
these two a plus a becomes 2 a and n
minus 3 plus 2 becomes n minus 1 n minus
1 see me all this becomes the same we'll
try the last 2 here also a plus n minus
2 D plus a plus D so a plus a becomes 2
a and n minus 2 plus 1 becomes n minus 1
n minus 1 D and this also becomes 2 a
plus n minus 1 so there are n number of
terms so we can say this is also equal
to n into 2 a plus n minus 1 D because
all these are 2 B plus n minus 1 D and
their end downs because this is 1 this
is 2 this is 3 this is 4 pi n comes so
we can say 2 of SN is equal to n into 2
plus n minus 1 D we divide both this
equation by 2 so what we get is SN is
equal to 2 N by 2 into 2 A plus n minus
1 into D and this we have proved so we
have proved using logic we have done
nothing we have
is equal to a a plus da plus 2 D then e
plus n minus 1 T also he will tell SL is
going to same thing in the reverse order
we have added is we have found that all
the sums are same that is 2 a plus n
minus 1 T so we don't waste is equal to
n into 2 n plus 20 dividing both the
equation by 2 we get s is equal to n by
2 into 3 plus n minus 1 D we take thank
you
visit exam viacom to watch free
education videos
try3 or my deaths get the best quality
study materials study from the best
tutors and mentors and much more
sale is part 5 is brought you by exam
fear calm no more fear from exam before
watching this video please make sure
that you have watched video part 1 -
part 4 let me introduce some optimally
in real life we have scenarios where you
need to find some up comes maybe example
let's suppose your teacher asked to
increase the number of questions you
solve per day by one who is ample today
you have solved one questions to morazán
proportions then three pushers and four
portions you keep increasing the calm
daily you increase by one and then you
want to find the number of questions you
have solved in the past hundred days so
it's different if your'e give you keep
adding one number one by one it is very
difficult to find answer because we have
got the formulas to find some of such
terms with easily the sum of terms Navy
is represented by n by 2 to a plus n
minus 1 D so let me tell you what are
the values n is the number of terms by
xanthine in this case you want to find
that 107 100 sum of 100 companies the
number of pushes we have solved in
hundred days then n will be 100 is the
first term so in this case would be 1
and D is the common difference in this
case it will also be 1 because every day
you are asked to increase your convent 1
also some of the terms in it we can also
become write n by 2 a plus L where L is
the last term's is I am well one the 579
and you know that this is till 179
this is a sequence of all odd-numbered
you know this value and you can find in
order to find it easier so you can them
say this is V and then by group do you
listen you similar to this because when
I'm saying n by 2 to M plus n minus 1 DS
I can
in this fashion and buy food cave less a
plus in Vegas 1d a plus n minus one D is
nothing but an extra what is n by 2 a
plus B and this is nothing but the small
same family I tried the mail the most
crucial formula is this so please
remember his formula some abdominal b is
nothing but n by Q into a plus n minus 1
D can he grooms formula for a week but
please commence nom nom some of them in
F is nothing but n by 2 into 2 M plus n
minus 1 D let's take for example we have
to find the sum of terms where a 3 6 9
12 15 za v first let's confirm where
this VPN on 3 6 minus 3 is 3 9 minus 6
is 3 12 minus 9 is 3 K minus 2 n is 3 so
it control that's a ap now I'm going to
find the number of terms 1 2 3 4 5 so
that means n is equal to 5
the first term is 3 is equal to see the
difference is 3 so D is the C so we have
M a and D we know that sum is nothing
but n by P into a Lance n minus 1 into D
so we can just fill in the values 5 by 2
into 2 into 3 plus 5 minus 1 is equal to
4 into D is equal to 3 so what you get
is 5 by 2 into 6 plus 8 6 plus 8 is 6
plus 12 this is 6 plus 12 is equal to e
teams are writing it in here and this
can sing this will 9 so 9 to 5 is going
to be 5 so sum of terms of this is 45
you can do a cross verification also
3+6 9 9 plus 9 18 18 plus 12 is 30 30
plus 15 is working 45 so when we do my
evaluation and we do the calculation
using formula what gives
that means our farmlands correct please
note some of the terms is n by 2 into
wave less n minus 1 D toggle proof
scholar using induction so formula says
sum of n term is equal to n by 2 to a
plus n minus 1 of D is the formula where
the sequence is of this form e e + D E +
- d e + 3 d continue is the sequence
what is the CDs consequence because
sequence is nothing but sum of series so
this sequence series nothing but sum of
sequence so this is sequence a a + D if
the story hitler's clearly now s/m is
going to invite to it we in the second -
1 D we have to prove by induction so to
prove by nation' what we do is we need
we as you we prove that this statement
is true for n is equal to 1 if it is
true for n squared - 1 we assume that
this statement is true for n is equal to
K and then we prove that statement is
true for n is equal to ki plus 1 so if
you can do these 3 steps that means
they're formalized consider to be ruled
by nature and is equal to 1 is true and
this vector K as you this it is true and
then we prove that it is true for air is
let you keep this one similarly we do
here s 1 is where n is equal to 1 we do
1 by 2 into into a plus 1 minus 1 is 0
into D so this becomes 1 by 2 into 2 8
and that becomes shape
so we see there our first immensity so
that means this is 2 now we'll assume
that s of K is 2 that is we will do that
as of K is equal to K by 2 into 2 a plus
K minus 1 into D then assume that this
is true now s of T plus 1
finger s of K plus T of n plus 1 or k
plus M because till K you have added
this plus reading T of T plus 1 all
right so this becomes T by 2 into 2 n
plus K minus 1 into D plus T of T plus 1
is nothing but a plus K plus 1 minus 1
into D plus 1 minus 1 castle that is a
plus K so if you solve this this becomes
K by 2 into 2 a this becomes ki a plus K
by 2 into K T this becomes K squared e
minus K by 2 into minus D this becomes K
2 D and this is plus a plus KD all right
so what we get here is what we get here
is he a plus K squared D by 2 this and
this we can add them scale D by 2 plus
KD by 2 plus a or we'll say we take a
common hand we'll put all the things you
see here this K a plus a plus we can say
K Square D by 2 plus KD by 2 so what we
get here is we take a common this
becomes T plus 1 into k plus 1 plus we
take here ki he D by 2 common into k
plus 1 this becomes nothing but he plus
1 into a plus KD by 2 this is the value
we got and this is nothing but s of K
plus one you can show you s of ki plus 1
using the formula it will become K by 2
k plus 1 by 2 into 2 e plus k plus 1
minus 1 into D that is gone this becomes
K plus 1 by 2 into so this becomes K
plus 1 or I can say k plus 1 into and
take to power here plus k dy so if you
see this and this what we have got our
same thus Y you can prove that this
formula is true using induction so
induction with induction we can certify
that this formula is true what we have
done we have proved that this is true
for n is equal to 1 if assume that this
is true for n is equal to K here and
then we have clue that this is true for
n is equal to k plus 1 thus we have to
the salvation formula that is SN is
equal to n by 2 into V Plus M minus 1 by
n minus 1 into D using that chunk now
let's try to prove the same thing using
logic so we'll write the sequence as you
can write the sequence as a this becomes
a plus D this becomes a plus 2 D a + 3 d
a plus holding dot dot dot a plus n
minus 2 d loves a plus n minus 1 because
SN is the last time is a plus n minus 1
T this is the sequence eviler also the
same sequence we can write like this in
the reverse order so let us write the
same sequence in the reverse order so
first we'll write a plus n minus 1 D
this will go me plus n minus 2 D correct
this here will write
plus n minus three here light a plus n
minus 40 this will become a plus n minus
5 D and this will become e this will
become eat less the same sequence I have
done in the reverse direction so you be
add these two you add these two this
becomes 2 SN this is equal to e plus a
plus n minus 1 D but you get a plus 2 a
plus n minus 1 B similarly e Plus D plus
a plus n minus 2 D this also becomes a
plus h 2 e M minus 2 plus 1 become n
minus 1 D similarly here also if you add
these two a plus a becomes 2 a and n
minus 3 plus 2 becomes n minus 1 n minus
1 see me all this becomes the same we'll
try the last 2 here also a plus n minus
2 D plus a plus D so a plus a becomes 2
a and n minus 2 plus 1 becomes n minus 1
n minus 1 D and this also becomes 2 a
plus n minus 1 so there are n number of
terms so we can say this is also equal
to n into 2 a plus n minus 1 D because
all these are 2 B plus n minus 1 D and
their end downs because this is 1 this
is 2 this is 3 this is 4 pi n comes so
we can say 2 of SN is equal to n into 2
plus n minus 1 D we divide both this
equation by 2 so what we get is SN is
equal to 2 N by 2 into 2 A plus n minus
1 into D and this we have proved so we
have proved using logic we have done
nothing we have
is equal to a a plus da plus 2 D then e
plus n minus 1 T also he will tell SL is
going to same thing in the reverse order
we have added is we have found that all
the sums are same that is 2 a plus n
minus 1 T so we don't waste is equal to
n into 2 n plus 20 dividing both the
equation by 2 we get s is equal to n by
2 into 3 plus n minus 1 D we take thank
you
visit exam viacom to watch free
education videos
try3 or my deaths get the best quality
study materials study from the best
tutors and mentors and much more
About this Video
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Apr 19, 2026 Last updated
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