** Binomial Theorem**

**Tell us what’s 2+3? **

The answer is of course 5.

Now, **tell us what’s (2+3)^2? **

We can do it in two ways:

1) Either we first calculate 2+3=5

and then 5^2 which is 25.

2) OR we can just apply the formula

of (a+b)^2 which __is__ a^2 + 2×a×b + b^2

Now you say that the first one was too simple and everyone will prefer to do it and of course it’s a right decision.

But now **tell us what’s 99^2? **

Now also we have two options, the one is by simple calculation and the other one is writing it as

(100-1)^2 and then applying the formula of (a+b)^2.

In this example, the second way is much better than the first.

**Why? **

Because here, our priority is accuracy and not time, and the second method will be easy to do so.

Now, if this method of solving problems is important, than we must know how the formula of (a+b)^2 came so that we may discover the formualas for higher powers also.

Let’s assume ourselves as ancient mathematicians or may be the ones who literally discover the formula of (a+b)^2.

Let’s try to observe the pattern first.

(2+3)^2 is 25.

Now 2^2 is 4 and 3^2 is 9.

And 4+9 is 13.It doesn’t matches the answer.

Now, 13+12 is 25.

So, we have to calculate 12 by using these two numbers.We can do it as

2×3=6 and then 2×6 which is 12.

So, we can write it as: Now, we may repeat the process for more examples and may get the formula of (a+b)^2.

Now, after getting the formula, our goal is to prove it. The proof of this formula is more than simple!! We can prove it as:

Now, after proving the above formula, we may want to find the formulas for other powers.

And by following the above steps we can easily find the formula for some other powers such as the formula for (a+b)^3 as

Now, it’s quite complicated to find the formula for every power and then remember it!!!

So, we have to talk about the general trend in these formula so we can easily find the formula for any power without performing complex calculations.

Let’s write down the formulas up to the power 4:

Now, if we observe the basic things in each formula, we may find that in each formula, the power of **a** starts from the the general power and keep decreasing till the last term where the power is 0.

Similarly, the power of **b** starts from 0 and keep increasing upto the highest one.

Now, let’s compare the coefficients as:

Hey!!

Look what we have!! It’s nothing but **Pascal’s triangle**. This is a triangle discovered by Pascal in which the top most element is 1 and then by taking the sum of the two elements, we can find the next term as shown below:

1+1=2, write it below

2+1=3, write it below

And so on!!

Now, we can easily find the next row of this triangle and can find the formula for (a+b)^5 by using the above observations.

Ok, so we have found a way to find the formula for any power.

But **what if we want to find the formula for (a+b)^9 or (a+b)^19 ?**

So we have to extend the triangle upto 9 or 19.

Hey!! this is really a lengthy and boring method!!

So, now our purpose is to find the desired row of Pascal’s triangle.

**How? **

Let’s look at the pattern again!!

If we see the marked portion,

We may write this as

1,3,6,10…..

Now, if we have to find the nth term of this series, we may notice that:

So, we can easily find the nth term by using the combination formula.

Now let’s look at the other marked portion:

Now, we have 1,4,10..so on. Again, if we want to find the nth term, we can do so as: Now, we are almost done!!!! Let’s write the Pascal’s triangle in terms of combination notation: Now, we can easily find any row of Pascal’s triangle as: If power is n, then the row is:

And hence we can easily find the formula for any higher power!!

Did you know this is nothing but **Binomial theorem **

Now, we may define the Binomial theorem as:

For any positive integer n,

We have: Now, if talk about the **general term**, for example if we want to find out the 4th term of the binomial expansion (a+b)^5, then we can easily found it as

So, here 3 is used commonly in the above formula and 3 is 4-1.But for finding the general term, we don’t write it as r-1 (for the rth term) because it will get more complicated. So, instead we write

The general term as

We have not denoted it by rth term but we have denoted it by( r+1) th term to make the expression simple.

So, if we want 5th term then r is 4 and we can easily apply it to the above formula.

**What if we want the middle term? **

Ok!! Let’s talk about it,

Suppose we have to find the middle term of 1,2,3,4,5,6,7.

Then, the middle term is 4.

Note that we have 7 terms which is an odd number and the middle term is 4th term and we can write 4 as (7+1)/2.So, **for any odd number of terms, we have the middle term as (n+1)/2.**

**What if we have even number of terms? **

Suppose, we have 1,2,3,4,5,6

Then, 3,4 are the two middle terms.

So, **we have two middle terms which are (n)/2 and (n)/2 + 1**

These are the two consecutive terms.

Now, in the binomial expansion **(a+b)^n**,

If n is even, then we have total n+1 terms which is odd.

**Note: **

In every binomial __expansion__

(a+b)^n, we have n+1 terms.

So now, the middle term is

(n+1+1)/2

Which is **(n/2 + 1)th** term.

Now, if n is odd, then n+1 is even and we have two terms which are

**1) (n+1)/2**

**And**

**2) (n+1)/2 +1**

these keywords are searched for our blog:

binomial theorem worksheet,

binomial expansion genetics,

binomial theorem examples,

binomial theorem proof,

binomial theorem calculator,

binomial theorem problems,

binomial theorem khan academy,

binomial theorem pdf,

Mononucleosis Amoxicillin Reaction Package Insert Amoxicillin Teva Pharaceuticals buy generic cialis Fluoxetine Panic Attacks Amoxicilina Next Day With Free Shipping

Shelf Life Of Zithromax Adult Dosing Amoxicillin levitra brand online Amoxicillin Dizziness

Levitra Come Acquistare Amoxicillin Penicillin Allergy Find Macrobid 100mg Bacterial Infections Visa viagra online Buy Synthroid No Prescription Order Kamagra Uk

Hi, letting you know that http://Get-Business-Funding-Now.com can find your business a SBA or private loan for $2,000 – $350K Without high credit or collateral.

Find Out how much you qualify for by clicking here:

http://Get-Business-Funding-Now.com

Minimum requirements include your company being established for at least a year and with current gross revenue of at least 120K. Eligibility and funding can be completed in as fast as 48hrs. Terms are personalized for each business so I suggest applying to find out exactly how much you can get on various terms.

This is a free service from a qualified lender and the approval will be based on the annual revenue of your business. These funds are Non-Restrictive, allowing you to spend the full amount in any way you require including business debt consolidation, hiring, marketing, or Absolutely Any Other expense.

If you need fast and easy business funding take a look at these programs now as there is limited availability:

http://Get-Business-Funding-Now.com

Have a great day,

The Get Business Funding Now Team

unsubscribe/remove – http://get-business-funding-now.com/r.php?url=boundtoteach.com&id=e164

Viagra Moins Cher France Propecia 1 Miligramo Zithromax Urethritis generic 5mg cialis best price Amoxicillin For Sale

http://mewkid.net/order-amoxicillin/ – Amoxil Children Amoxicillin Online mor.kdga.boundtoteach.com.ioz.oe http://mewkid.net/order-amoxicillin/

http://mewkid.net/order-amoxicillin/ – Amoxicillin Buy Amoxicillin teu.yoiv.boundtoteach.com.nff.jj http://mewkid.net/order-amoxicillin/

Faster and Easier than the SBA, http://SmallBusinessLoans24-7.com can get your business a loan for $2K-350,000 With low-credit and without collateral.

Use our short form to See exactly how much you can get, No-Cost:

http://SmallBusinessLoans24-7.com

If you’ve been in business for at least one year you are already pre-qualified. Our Fast service means funding can be finished within 48hrs. Terms are personalized for each business so I suggest applying to find out exactly how much you can get.

This is a free service from a qualified lender and the approval will be based on the annual revenue of your business. Funds have no Restrictions, allowing you to use the full amount in any way including bills, taxes, hiring, marketing, expansion, or Absolutely Any Other expense.

There are limited SBA and private funds available so please apply now if interested,

Click Here: http://SmallBusinessLoans24-7.com

Have a great day,

The Small Business Loans 247 Team

remove here – http://smallbusinessloans24-7.com/r.php?url=boundtoteach.com&id=e165

A nnw.spws.boundtoteach.com.pnk.ma polyostotic [URL=http://circulateindia.com/cialis/ – cialis[/URL – [URL=http://pccarebusiness.com/propranolol/ – cheapest propranolol[/URL – [URL=http://theatreghost.com/isoniazid/ – isoniazid[/URL – [URL=http://gaiaenergysystems.com/buy-viagra/ – viagra online[/URL – viagra canada [URL=http://antonioscollegestation.com/alli/ – alli lowest price[/URL – buy alli [URL=http://circulateindia.com/cialis-20-mg/ – cialis[/URL – [URL=http://hackingdiabetes.org/depakote/ – depakote[/URL – [URL=http://casatheodoro.com/amoxicillin/ – amoxicillin 500mg[/URL – dislocation: cialis online propranolol isoniazid lowest price viagra buy viagra alli online generic cialis lowest price depakote pills amoxicillin 500mg capsules imperfect, cigarette http://circulateindia.com/cialis/ generic cialis in canada low cost cialis http://pccarebusiness.com/propranolol/ propranolol in hyperthyroidism http://theatreghost.com/isoniazid/ isoniazid online http://gaiaenergysystems.com/buy-viagra/ viagra http://antonioscollegestation.com/alli/ alli online http://circulateindia.com/cialis-20-mg/ generic cialis online http://hackingdiabetes.org/depakote/ order depakote online http://casatheodoro.com/amoxicillin/ amoxicillin 500mg capsules counsellors dosage extinguished.

Pregnancy uhe.hadl.boundtoteach.com.okz.ah antivirals, [URL=http://vintagepowderpuff.com/pharmacy/ – canadian pharmacy[/URL – [URL=http://elegantearthatthearbor.com/mobic/ – buy mobic[/URL – [URL=http://gormangreen.com/prometrium/ – cheapest prometrium[/URL – [URL=http://bikramcentennial.com/eulexin/ – eulexin canada[/URL – [URL=http://recipiy.com/diclofenac-gel/ – diclofenac gel[/URL – [URL=http://bikramcentennial.com/altace/ – online altace[/URL – [URL=http://csharp-eval.com/nolvadex/ – buy nolvadex[/URL – [URL=http://cheapflights-advice.org/lyrica/ – lyrica canada[/URL – [URL=http://elegantearthatthearbor.com/altace/ – altace online[/URL – intake; canadian pharmacy cialis 20mg canadian pharmacy online buy mobic prometrium without a prescription eulexin online diclofenac gel for sale altace for sale nolvadex for sale nolvadex lyrica altace pituitary, practitioner http://vintagepowderpuff.com/pharmacy/ canadian pharmacy online http://elegantearthatthearbor.com/mobic/ order mobic online http://gormangreen.com/prometrium/ price of prometrium http://bikramcentennial.com/eulexin/ eulexin pills eulexin http://recipiy.com/diclofenac-gel/ diclofenac gel for sale http://bikramcentennial.com/altace/ altace http://csharp-eval.com/nolvadex/ nolvadex http://cheapflights-advice.org/lyrica/ cheap lyrica http://elegantearthatthearbor.com/altace/ altace is: booking, therapies.