Arithemetic Sequence

Question

A small company producing children’s toys plans an increase in output.
The number of toys produced is to be increased by 8 each week until the
weekly number produced reaches 1000. In week 1, the number to be
produced is 280; in week 2, the number is 288; etc. Show that the weekly
number produced will be 1000 in week 91.
---From week 91 onwards, the number produced each week is to remain at
1000. Find the total number of toys to be produced over the first 104 weeks
of the plan.

Answer 1

280 + 8(n-1) is the weekly production number for week n.

If n=91, then this production is 280 + 8(90) = 280 + 720 = 1000.

 

The total produced under this scheme, in n weeks, is S_n = n/2(a₁+ a_n).

Substitute n = 91, a₁ = 280, a_n = 1000 to get 91-week production total (91/2)(280+1000) = 91(640) = 58,240.

 

For weeks 92 through 104 (13 additional weeks), there will be production total of 13,000.

 

Check this work!  Then add the two underlined numbers to get the total number of toys to be produced over the first 104 weeks of the plan.