Question

Suppose the singers true love gave presents for n days instead of 12 days what would the toatl number of gifts given on the nth day be?she told us that the sum of consecutive integers from 1 to n is n(n+1) over 2
Someone told me n= 12 and the sum is = to 78. Why?

Answer 1

G(n) = 1 + 2 + 3 + 4 + 5 + ....n
to get the average of that sequence add the first and the last and divide by 2
for example
1 + 2 + 3 + 4 + 5 + 6 = 1+6 + 2+5 + 3+4
= 7 + 7 + 7
the average is 7 which is 1+6 divided by 2 (or 2+5 divided by 2 or 3+4 divided by 2
)
then multiply the average (1/2)(n+1)
by the number of terms, n, to get the total.
So yes, your formula is right.
1 + 2 + .... 12
= (1/2)(1+12)(12)
=13*6
= 78 sure enough

Answer 2

I explained that poorly:

G(n) = 1 + 2 + 3 + 4 + 5 + ....n
to get the sums of that sequence add the first and the last and the second and the second to last etc. in pairs
1 + 2 + 3 + 4 + 5 + 6 = 1+6 + 2+5 + 3+4
= 7 + 7 + 7
each sum is 7 which is 1+6 divided by 2 (or 2+5 divided by 2 or 3+4 divided by 2
)
and there are n/2 of those pairs
So yes, your formula is right.
1 + 2 + .... 12
= (1/2)(1+12)(12)
=13*6
= 78 sure enough

Answer 3

aaarrgggh another mistake

G(n) = 1 + 2 + 3 + 4 + 5 + ....n
to get the sums of that sequence add the first and the last and the second and the second to last etc. in pairs
1 + 2 + 3 + 4 + 5 + 6 = 1+6 + 2+5 + 3+4
= 7 + 7 + 7
each sum is 7 which is 1+6 (or 2+5 or 3+4 )
and there are n/2 of those pairs
So yes, your formula is right.
1 + 2 + .... 12
= (1/2)(1+12)(12)
=13*6
= 78 sure enough