Word problem

Question

Tommy has taken a job in which he earns $1 on the first day of the month, $2 on the second day, $3 on the third, and so on. How much money will he earn in the month of October with 31 days?

Answer 1

1+2+3+4+5+...+27+28+29+30+31=???

add the first and the last

add the second and the next to last

add the third and the next to the next to the last and so on

1+31=32

2+30=32

3+29=32

4+28=32

5+27=32

there are 15 of these sums and one lone number in the middle, the number 16

1-15, 16, 17-31

15*32=480, 480+16=496

 

another solution...

the sum of an arithmetic sequence is given by the formula...

S(n)=(n/2)[2a+(n-1)d] where n=31, a=1(first term), and d=1(difference between terms)

S(n)=(31/2)[2*1+(31-1)1]

S(n)=(31/2)(2+30)

S(n)=(31/2)(32)=(31)(16)=496

Answer 2

the sum of positive integers from 1 to n equals n(n+1)/2

 

in this case n = 31, therefore we get 31(32)/2 = 496