Question

A mother is convinced by her son that he should have a weekly allowance that is doubled every 2 weeks. In weeks 1 and 2 he receives 10¢ per week; in weeks 3 and 4 he receives 20¢ per week. If his allowance continues to double every 2 weeks, what is the amount of his allowance in week 30 (rounded to the nearest cent) and what is the total amount he receives in allowance for the 30 weeks (rounded to the nearest cent)?

A) The son receives $3,276.80 in week 30 and he accumulates $6,553.40 in total for the 30 weeks.
B) The son receives $3,276.80 in week 30 and he accumulates $65,534.00 in total for the 30 weeks.
C) The son receives $1,638.40 in week 30 and he accumulates $6,553.40 in total for the 30 weeks.
D) The son receives $1,638.40 in week 30 and he accumulates $3,276.70 in total for the 30 weeks.

Answer 1

so hmmm every two weeks, he'll get 10cents,

week 2 is 10cents
week 4 is 20 cents
week 6 40

and so on.

so, we know for the first week pair he'll be getting 10cents, so let's say the each term in this sequence is a week pair and since there'll 30 weeks, then that means 30/2 or 15 week pairs or 15 terms.

now, the first term is 10, and it doubles from there on, thus the "common ratio" is 2.


`\bf n^(th)\textit{ term of a geometric sequence}\\\\ a_n=a_1\cdot r^(n-1)\qquad \begin{cases} n=n^(th)\ term\\ a_1=\textit{first term's value}\\ r=\textit{common ratio}\\ ----------\\ a_1=10\\ r=2\\ n=\stackrel{week~pair}{15} \end{cases} \\\\\\ a_(15)=10\cdot 2^(15-1)\implies a_(15)=10\cdot 2^(14) \\\\\\ a_(15)=10\cdot 16384\implies a_(15)=163840~cents`

now, the 15th term, or 30th week, it'll be 163840 pennies, how many dollars is that? well, 163840/100 or 1638.40.