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(1)- Find the sum of 1+3+7... _ 57 (nth odd is (2n-1)

(2)- Find the sum of the first 28 counting numbers. 

 Feb 19, 2020
 #1
avatar+109740 
+1

(1)  1 + 3 + 5 + ....+  57

 

Note that the sum of the first 2 odd positive integers  = 2^2

And the sum of the  first  3 positive odds = 3^2

And 

2n  -1 =  57

2n = 58

n = 29

So 57 is the 29th positive odd

 

So.....the sum of  1 +3 + 5 + ....+ 57   is the sum of the first   29th odd numbers = 29^2  = 841

 

 

(2)  Sum of 1st n counting numbers =  n (n + 1)   / 2

 

So....the sum of the  first 28 counting numbers =   28 * 29 / 2  =  14 * 29  = 406

 

 

cool cool cool

 Feb 19, 2020
 #2
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(1)- Find the sum of the first 28 odd couting numbers

 

(2)- Develop the formula of the sum of the first n counting numbers. 

S= 1     +    2    +   3    + ... + (n-2) +   (n-1) + n

 

(3)- Find the sum of the first 499 counting numbers 

 Feb 19, 2020
 #3
avatar+109740 
+1

1.     From above  → 28^2   = 784

 

2. Sum of  first  n counting numbers

 

Note  that  we  have

 

1 + 2 + 3 + 4 + .....+  n - 3 + n -2  + n -1  + n

 

Note that we can rearrange this as

 

(1+ n) + ( n  -1 + 2)  + ( n - 2 + 3)  + ( n - 3 + 4) + .....  =

 

(n + 1) + ( n + 1)  + (n + 1)  + ( n + 1)  + ....

 

We have  n /2  of these pairs....so.....the sum is just     n * ( n + 1)  / 2

 

3. Sum of 1st 499 counting numbers =   499 * ( 500) /2  = 499 * 250   = 124,750

 

 

cool cool cool

CPhill  Feb 19, 2020

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