Find out the sum of the following serie:

Question

5+9+13+17+...+89

Answer 1

The pattern in your sequence is 4n+1 Hence, 

 

4n+1=89

      -1   -1

 

4n=88  n=22 terms

 

The way to derive the formula for the sum is to add the first term and the last term which becomes

 

5+4n+1=4n+6

 

Add the second term and second to last term in which becomes

 

9+4(n-1)+1=9+4n-4+1=4n+6

 

Hence the sum of the series would be half the number of sequences times 4n+6. Therefore, the formula becomes:

 

S(n)=n/2*(4n+6)

 

For n=22

 

S(22)=22/2*(4*22+6)=11*(88+6)=11*(94)=1034

 

Thus, the sum of the series is 1034.

Answer 2

5+9+13+17+...+77+81+85+89

First find out how many numbers there are.

If you had only 5+9+13, you would have only 3 numbers.

13-5=8, 8/4=2, and 2+1=3 (The common difference is 4)

If you had 5+9+13+17 you would have only 4 numbers.

17-5=12, 12/4=3, and 3+1=4

89-5=84, 84/4=21, and 21+1=22 numbers !!

add the first and the last: 5+89=94

add the second and the next to last: 9+85=94

add the third and the next to next to last: 13+81=94

there are 22 numbers and you are grouping them by 2's

therefore there are 11 groups and each group sums to 94

11*94=1034 is the sum of the terms of the sequence