Proof Without Words - Part 3

Algebra Level 1

Which of the following is demonstrated by the diagram above:

( a + b ) ( a − b ) = a 2 − b 2 (a + b)(a-b) = a^2 - b^2 1 2 + 2 2 + 3 2 + ⋯ + n 2 = n ( n + 1 ) ( 2 n + 1 ) 6 1^2 + 2^2 + 3^2 + \dots + n^2 = \dfrac{n(n+1)(2n+1)}{6} 1 + 3 + 5 + ⋯ + ( 2 n − 1 ) = n 2 1 + 3 + 5 + \dots + (2n-1) = n^2 1 + 2 + 3 + ⋯ + n = n ( n + 1 ) 2 1 + 2 + 3 + \dots + n = \dfrac{n(n+1)}{2}

Arron Kau by Arron Kau

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2 solutions

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Prasun Biswas
Mar 2, 2014

You can see in the diagram that on the lower right corner, there is one ball in the darkly coloured region (say shell 1) and as we go on to each successively lighter coloured regions (say shells 2,3,...,n), the no. of balls in the p p th shell can be expressed in the form ( 2 p − 1 ) (2p-1) .

Like, shell 1 1 has ( ( 2 × 1 ) − 1 ) = ( 2 − 1 ) = 1 ((2\times1) -1)=(2-1)=1 ball,

Shell 2 2 has ( ( 2 × 2 ) − 1 ) = ( 4 − 1 ) = 3 ((2\times2) -1)= (4-1)=3 balls,

Shell 3 3 has ( ( 2 × 3 ) − 1 ) = ( 6 − 1 ) = 5 ((2\times3) -1)=(6-1)=5 balls , and

Shell n n has ( 2 n − 1 ) (2n-1) balls.

Let there be n n shells starting from 1 1 to n n . Let the total no. of balls enclosed collectively from the 1st shell till n n th shell be considered. Then, from the diagram, we see ----

Total balls till 1 1 st shell = 1 = 1 2 =1=1^2 ,

Total balls till 2 2 nd shell = 1 + 3 = 4 = 2 2 =1+3=4=2^2 , . . . . . . . ,....... and till n n th shell = 1 + 3 + . . . . + ( 2 n − 1 ) = n 2 =1+3+....+(2n-1)=n^2

So, we see that this diagram represents the following ----

1 + 3 + 5 + . . . . . . . + ( 2 n − 1 ) = n 2 \boxed{1+3+5+.......+(2n-1)=n^2}

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We can see that it's an arithmetic progression. We can see that each next units are gotten by adding up the difference, not by multiplying it.

1, 3, 5, 7, ... 2 is the distance of the sequence.

So, the formula for this sequence is:

a n a_n = a 1 a_1 + (n-1)d

a n a_n = 1 + (n-1)2

a n a_n = 1 + 2n - 2

a n a_n = 2n - 1

But, we were asked the sum of that sequence. So, we use the arithmetic series formula:

S n S_n = n ( 2 a 1 + ( n − 1 ) d ) 2 \frac {n(2 a_1 + (n-1)d)}{2}

S n S_n = n ( a 1 + a n ) 2 \frac {n (a_1 + a_n)}{2}

S n S_n = n ( 1 + 2 n − 1 ) 2 \frac {n (1 + 2n - 1)}{2}

S n S_n = n ( 2 n ) 2 \frac {n (2n)}{2} = n 2 {n^{2}} .

That's it!

Eka Kurniawan - 7 years, 3 months ago

thanks, i don't know but i am feeling shame on me coz I don't get that this was an ap question, i just though if it was mathematical induction.

Ayush Porwal - 7 years, 2 months ago

i understand (1+3+5...................+2n-1 ) eqn. but how came n2 ????

Lalita Verma - 7 years, 3 months ago

You need to have knowledge about Arithmetic Progression in order to understand this. Refer to this link to get an understanding on AP.

You should know that for sum of an AP, you have the formula S = n 2 { 2 a + ( n − 1 ) d } = n 2 ( a + l ) S=\frac{n}{2}\{2a+(n-1)d\}=\frac{n}{2}(a+l) where a = a= First term of AP, d = d= Common difference of AP, n = n= No. of terms in AP and, l = l= Last term of AP.

We can see that the sequence 1 , 3 , 5 , . . . . . , ( 2 n − 1 ) 1,3,5,.....,(2n-1) is an AP with a = 1 , d = 2 and l = ( 2 n − 1 ) a=1,d=2 \text{ and } l=(2n-1) .

So, the sum of this series = n 2 ( a + l ) = n 2 ( 1 + 2 n − 1 ) = n 2 × 2 n = n × n = n 2 =\frac{n}{2}(a+l)=\frac{n}{2}(1+2n-1)=\frac{n}{2}\times 2n = n\times n = \boxed{n^2}

Reply if you have any further doubts.

Prasun Biswas - 7 years, 2 months ago

To arrive at the n 2 n^2 part of the equation, look at the general form: 2 n − 1 2n - 1 .

n = 1 ⇒ 2 ( 1 ) − 1 = 2 − 1 = 1 n = 1 \Rightarrow 2(1) - 1 = 2 - 1= 1

n = 2 ⇒ 2 ( 2 ) − 1 = 4 − 1 = 3 n = 2 \Rightarrow 2(2) - 1 = 4 - 1 = 3

n = 3 ⇒ 2 ( 3 ) − 1 = 6 − 1 = 5 n = 3 \Rightarrow 2(3) - 1 = 6 - 1 = 5

and so on.

You can see from the illustration that 1 + 3 = 4 1 + 3 = 4 , which is basically 2 2 2^2 .

The same goes with the next one: 1 + 3 + 5 = 9 = 3 3 1 + 3 + 5 = 9 = 3^3 .

Lanvin De Los Santos - 7 years, 2 months ago

can you prove this statement

vaishnav garg - 7 years, 3 months ago

I think I did it in my solution !! Do you mean if I can prove it mathematically and without any diagram ??

Prasun Biswas - 7 years, 3 months ago

Let n=7, than 1+2+3+4+5+6+7=28 than 28=7(7+1)/2, now 28=7(8)/2, now 28=56/2, hence 28=28

hence, the correct answer is 1+2+3....n= n(n+1)/2

Muhammad Durrani - 7 years, 3 months ago

No, actually I dont remember the options of this question but most of the options are true formulas but here it is asked which formula is being represented by the diagram given and from the diagram, we see that it is the representation of the formula 1 + 3 + 5 + . . . . . + ( 2 n − 1 ) = n 2 1+3+5+.....+(2n-1)=n^2 and not 1 + 2 + 3 + . . . . + n = n ( n + 1 ) 2 1+2+3+....+n=\frac{n(n+1)}{2}

Prasun Biswas - 7 years, 3 months ago

hey, please notice that the list is 1,3,5,7,... not 1,2,3,4,...

Eka Kurniawan - 7 years, 3 months ago

There is one ball in the 1st shell so 1=1;3 balls in 2nd shell so total no of balls upto 2nd shell is 1+3=4=2^2; 5 balls in the 3rd shell so total no balls upto 3rd shell is 1+3+5=9=3^2; Generalizing this equation upto (2n-1)th odd term as no of increasing balls in each shell is odd we get 1+3+5+.............+(2n-1)=n^2.so this is the equation which demonstrate the diagram exactly.

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