Bobby and Emily each got $100 red packet money from their grandmother when they were born. Then, their grandmother gave them red packet money on each of their birthdays. The amount of money given to Bobby was $50 more than the previous year, while the amount given to Emily was 20% more than the previous year.
(a) How much did each of them get when they were 8 years old?
--->Bobby=$500; Emily=$????
(b) Find the total amount of red packet money that each of them got until they were 8 years old.
--->Bobby=$2700; Emily=$????
(c) Emily claims that she will get more packet money than Bobby when they are both 10 years old.
(i) is Emily's claim correct? Why?
(ii) Find the difference between their total red packet money when they are 10 years old.
ok, the first part is just an arithmetic sequence,
for bobby, (100,150,200,250....)
so, first term = a1 = 100 and common difference = 50
9th term (after 8 years) = a1 + (n-1)d = 100 + 8*50 = $500 (correct)
but for emily :
(100 + 100*0.2 +100*0.2*0.2+...)
= 100 (1+0.2 +0.2^2+0.2^3+...)
now this is geometric sequence
a1 = 1, r = 0.2
so, for 9th term (after 8 years)
= a1 (r)^(n-1) = 1 * (0.2)^8 =.... ?
multiply the result by 100 to get the answer...
for bobby, Arithmetic sequence
n'th term formula
\(T_n = a_1+(n-1)d\)
sum formula
\(S_n = (n/2)[2a_1+(n-1)d]\)
for emily , geometric sequence :
n'th term formula \(T_n = a_1r^{n-1}\)
sum formula \(S_n = a_1 [\dfrac{r^n-1}{r-1}]\)
for emily, the sequence will be
100 , 100*0.2 , 100*0.2*0.2 , .... any doubts in this ?
so, taking 100 out,
100 ( 1, 0.2, 0.2*0.2,....)
in the above sequence first term = a1 = 1
but for emily :
(100 + 100*0.2 +100*0.2*0.2+...)
now this is geometric sequence a1 = 100, r = 0.2
so, for 9th term (after 8 years) = a1 (r)^(n-1) = 100 * (0.2)^8 =.... ?
ok, its like compound interest formula!
let me revise the formula
n'th term
\(T_n = a_1 (1+r)^{n-1}\)
here, a1 = 100, r=0.2, n = 9 gives
T9 = 100 (1.2)^8 =.... ?
i want to know , in part c, the statement said "Emily claims that she will get more red packet money than Bobby when they are both 10 years old", total amount or the money in that year?
for bobby
first term = a1 = 100 and common difference = 50
11th term (after 10 years) = a1 + (n-1)d = 100 + 10*50 =600 (correct )
T11 = 100 * (1.2)^10 = 619.174 (correct)
and YES , emily's claim was correct :)