Hello everyone, welcome back to another video.
In the last video, just recent previous video, we did a new
topic that was Arithmetic Progression.
Some questions were left out of it. I said that we will do
it in the next video.
This is the next video.
So let's see what were those questions in it.
I hope you will remember those two formulas well.
AN is equal to A plus N minus 1 D.
You have done the sum of the nth term.
N by 2.
A plus L.
These two are done.
And one more thing, I had told you the result of this.
What should be the sum of first and national number?
What is it?
N into N plus 1 divided by 2.
So let's see.
Let's warm up.
You can do it quickly with this question.
If you have understood the last question well.
The third term of an AP is 18.
How will you denote the third term?
From A3 which is 18.
And the seventh term is 30.
Find the sum of 25 terms.
In this, the value of N is given.
In the rest, you had to subtract the value of N.
In this, the value of N is given.
You have to find the sum of 25 terms.
So, A3 is given.
A7 is given.
N is given.
If you don't have given, then what is not given?
The value of A is not given.
And the value of L is not given.
Or the value of D.
If you are not able to find the value of L.
Then either use the second formula.
N by 2.
2A plus N minus 1 D.
Either you will get the value of D.
Or you will get the value of A.
So, it is up to you.
For the value of A and D, I had told you.
To find both, you will be given two things.
A3 is given.
And A7 is given.
So, what do we put N first?
First, we put N as 3.
So, you will get A3 in the formula of AN.
A3 is equal to A plus 2D.
N minus 1 is equal to 2D.
So, you will get 18.
First equation is this.
Now, putting N is equal to 7.
If we put N as 7, then A7 is equal to A plus 6D.
N minus 1 is equal to 6D.
Which is 30.
Solve both.
How will you do?
Subtract.
Then, you will get minus sign.
A minus A is equal to 0.
2D minus 6D.
Minus 4D.
18 minus 30.
That is minus 12.
D minus is equal to minus.
Plus is equal to plus.
Cancel is equal to 12 by 4.
D is equal to 3.
When D is equal to 3,
Put D in any equation.
You will get the same value.
Suppose, we did it in the first equation.
2 into 3 is equal to 6.
A plus 6 is equal to 18.
What is the value of A? 18 minus 6 is equal to 12.
We got A and D.
N was already given.
Put N by 2 in the formula.
That is 25 by 2.
We have to find the formula for the sum.
What will we write instead of N? 25.
We have to sum up to 25.
S25 is equal to 25 by 2.
Into A is equal to 12.
We are not doing the formula for L.
Multiply 12 by 2.
N minus 1 is equal to D.
25 minus 1 is equal to 24.
D is equal to 3.
Will you be able to solve this equation?
To solve this equation,
You will get the final answer.
25 by 2 is equal to 24.
Plus 12 is equal to 1.
6 is equal to 72.
So, 6 and 92.
96 divided by 2 is equal to 25.
Multiply all the numbers.
48 into 25 is equal to 25.
That would be the sum of 25 terms of AP.
This is the warm up of the last class.
I told you that we will do some word problems.
See the word problems.
Read the question.
How will we convert the word problems into AP?
We will convert them into AP.
We will solve them using the APK formula.
A man eats 73 kiwis in 5 days.
Each day eating 3 more kiwis than the previous day.
In 5 days.
You might have 5 days.
1st day, 2nd day, 3rd day, 4th day and 5th day.
He has eaten 73 kiwis in 5 days.
How many? 73 kiwis.
On every new day.
Suppose he ate A on 1st day.
Let's take A.
How many kiwis did he eat on 2nd day?
More than 3 kiwis.
He is eating 3 extra kiwis every day.
What will happen on 2nd day?
On 3rd day.
On 2nd day, he ate 3 extra kiwis.
3 plus 3 is 6.
So, A plus 6.
Similarly, on 4th day.
On 5th day,
He will eat A plus 12 kiwis.
These are the number of kiwis he has eaten in 5 days.
The addition of all these numbers should be equal to 73.
73 is equal to A.
It is very simple.
Add A plus A plus 3.
1st term, 2nd term, 3rd term, 4th term and 5th term.
Add all the kiwis he ate in 5 days.
It is equal to 73.
Now, you have only one unknown variable.
That is A.
You have only one equation.
You will solve it.
You will find the value of A.
What will happen?
How many times is A? 1, 2, 3, 4, 5.
5 times A.
Add 3 plus 6.
9 plus 9 is equal to 18.
18 plus 12 is equal to 30.
What will happen? 9 plus 9 is equal to 18.
18 plus 12 is equal to 30.
30 is the answer.
You will subtract 30.
Sorry, this is a mistake.
This is not 5 days.
This is 6 days.
You have the data of 6 days.
Okay? He ate this in 6 days.
Even if you find the answer in 5 days, do not change the
question.
If you solve the question in 5 days, we will take the
answer in 5 days.
We will solve it in points.
It will not be complete.
What will be the answer? 73 minus 30.
That will be 43.
What will you do?
You will subtract one value.
What will be the value of A? 43 divided by 5.
The answer should have come in the whole.
But it is not coming in the whole.
You do not have any problem.
Because the question is like that.
As long as you are going on the right path, the answer is
coming in the point or the whole.
You do not have to worry.
You have to write your final answer.
You have to mark the answer by tallying the options.
You have to find the value of A.
How much did he eat in the first day?
He must have eaten A.
That will be 43 divided by 5.
Okay? If we had solved this, what would have happened?
Instead of 73, if it was 70 or 75, which is divisible by 5,
then this question would have been solved.
It would have come in the whole number.
But this is also right.
Because we have chosen the right path.
Okay? So, the word problems are clear.
How to convert them into AP?
Then we will apply the AP formula and solve the rest of the
questions.
Now, see this.
This is another interesting question.
You have to keep this pattern in mind.
If you keep this in mind, then it will be easily solved.
I will tell you the answer.
The sum of three numbers in AP is 24.
And their product is 440.
These types of questions are frequently asked.
Find the numbers.
You have to tell the three numbers.
Okay? Now, in these numbers, it is not necessary that you
have to write the sum of three numbers.
You can also write the sum of consecutive three numbers.
Even then, what will be the difference?
In that, you will not mention AP.
But you have to keep in mind that the sum of three
consecutive numbers is 24.
natural number.
So they will form AP because there is a gap of 111.
So now they have directly written that it is AP only, so we
don't have any problem.
So sum of three numbers in AP, whenever you have a question
of three numbers in AP, the three numbers you will take,
first three numbers A1, A2, A3, instead of A, A plus D, A
plus 2D, instead of taking these, you can take these also,
just the calculation
will be little more, nothing else, you can take these also
for sure, but instead of taking these, we prefer to take
this, A minus D, go one step back from this, A minus D is
done, then A as it is and A plus D.
Okay? Here, we have taken the third term to move forward,
we have taken the third term to move backward, so those
three numbers.
So always remember this, whenever you are asked about three
numbers in AP, then you take this combination, it will be
solved quickly, it will be solved easily rather than this.
This will also be solved, but it will take you a little
more time, this
A minus D, A, A plus D, let these be three numbers of AP,
in AP.
There are three terms of AP, now their sum, according to
question, what is their sum given to you? A minus D plus A
plus A plus D, their sum is given to you as 24.
Now see, why did I take these numbers?
Because, as soon as we keep this equation, D to D will get
cancelled out.
If you would have done it here, then here D plus 2D would
have become 3D, then you would have A also and along with
it, D would also remain in the equation.
In this, you have D cancelled out, so you will get the
value of A directly, 3A is equal to 24, which implies A is
equal to 8.
You have got the value of A, 8.
Now, second thing you have to ask is the product.
Their product is given to you as 440.
What will be their product? A minus D into D into, sorry
not D, A, A into A plus D.
Their product should be equal to what? Equal to 440.
Now, you have taken out the value of A first, 8, so
substitute it there.
So, 8 minus D, 8, 8 plus
D is equal to 440.
Divide 8 here, so 8 minus D, 8 plus D is equal to 440
divided by 8, that is 45, 55.
Now, you either multiply these two brackets by 8, then
minus D.
Either you put identity in it, A minus B.
If you have two numbers, they are getting added and getting
subtracted.
A minus B into A plus B.
What is their equal? I will write A minus
B into A plus B.
One time plus and one time minus.
So, this is equal to A square minus
B square.
So, either you use this or simply multiply them.
There is no problem.
I am using this identity.
So, A, 8, 8 square minus D square is equal to 55.
So, 8 square is equal to 64.
64 minus D square is equal to 55.
Take 55 here and D square here.
So, 64 minus 55 is equal to 9.
9 is equal to D square.
Under root, you will get two values of D plus 3.
How many values of D are there? Two values, plus 3 and
minus 3.
So, you will have two cases.
One you will have, I will erase this.
One you will have case of plus 3 and one you will have case
of minus 3.
So, you will have two set of answers in this.
So, keep value of A as 8 and keep value of D as 3 and keep
value of D as 3 and keep value of A as 2.
So, value of A will remain same and value of D will be
equal to minus 3.
So, what is the first term? A minus D, that is 8 minus 3.
8 minus 3 is 5.
Second term A, that is 8.
Third term A plus D, 8 plus 3, that is 8 plus 3 is 12.
So, 5 comma 8 comma 12.
What is this? This is three numbers of AP, whose sum will
be 24 and product is 440.
So, do the same case in this, in minus 3.
This is your homework.
What will you do? You have to put value of minus 3 instead
of D.
So, what will be the first? 8 minus of minus 3, that is 8
plus 3 is 12.
Then, second A, that is 8.
Then, third A plus D, that is 8 plus minus 3, that is 8
minus 3 is 5.
So, numbers are same, 5, 8, 12, but their order is
different.
This 5, 8, 12 is in increasing order and this 12, 8, 5 is
in decreasing order.
So, this is your homework.
When you have three numbers of AP, whose sum and product,
then which combination you have to take, keep this in mind.
Moving on to last question of this video.
Read it.
First of all, I will change you to AP and you will solve it
once from home.
it fast.
We have word problem.
A man gets employment of Rs.
750 per month.
How much rupees is he getting? Rs.
750 per month with an annual increment of annual.
Keep in mind
annual increment of Rs.
50. Means, yearly.
How much is his increment of Rs.
50 in one year? Then, what does he earn in 10 years? So,
how much is he earning in 10 years? First of all, this is
very less salary given.
Increment is also less.
But, this is given by the tone
of the person.
So, your income per month is Rs.
750. So, what will be your income in one year? If you
multiply Rs.
750 by 12, then your per annum, which you call as package,
will be your salary.
So, first of all, you calculate salary per annum or salary
in one year.
How much salary will be there in one year? Rs.
750 which was of one month, if you multiply it by 12, then
your income in one year will be Rs.
750. So, Rs.
9000. Rs.
9000 is what he is earning annually.
Now, as he worked for one year, his first year salary, he
got Rs.
9000. As he grew up in the second year, he got an increment
of Rs.
50 per annum.
So, how much money will he get in the second year? Rs.
9000 plus Rs.
50 increment.
salary will be Rs.
9050 for the next month.
So, his total income in the second year will be Rs.
9050. This is his income in the second year.
Third year income, third year income. In the third year, Rs.
50 will be added again.
Rs. 9050 plus Rs.
50. So, total Rs.
10,000 will be there in his third year.
Similarly, his annual income is increasing.
Rs. 50, Rs.
50. Increment is happening.
What you have to see is that how much he is earning in the
second year.
So, you can calculate the annual data.
AP form is formed.
First, Rs.
9000, then Rs.
9050, then Rs.
10,000, then Rs.
10,050 and so on.
This will continue for 10 years.
And you have to find the sum of all these.
So, you will get the formula for sum.
What is the value of n? You will take the value of n as 10.
Write the formula for sum.
Sum formula will be SN is equal to N by 2, 2A plus N minus
1 D.
D you have 50, N you have 10 years, so 10 by 2, 2A, A is
9000, so 9000 plus N minus 1, that is 10 minus 1 is 9 and D
is increment 50, 9 into 50.
You will solve this, it will become 5.
This will become 18000 plus 9 into 5, 450.
So 18000 plus 450, 18450.
You multiply this by 5, you will get the final answer.
In 10 years, how much was his total salary? It was,
whatever it was, whatever he used.
So multiply this by 5, you will get 25, then 20, 22, 40, 42.
The total income he will have, that is, is it less? 9, 2,
2, 5, 0.
18,000, 92,250 rupees.
That is the total salary saving of 10 years, which you had
to find out.
I hope all these problems that we did yesterday, today,
whatever formulas we have done, all of them would have come
out well and the concept of AAP would have become very
clear. It will become more clear when we do GP and do mixed
questions of GP and
AAP.
So till then you practice AAP and revise the old topics
again and again.
Thank you for watching the video.
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