Method
{"text":"bold","word":"The Approach"}
Add the previous two numbers to get the third term of the sequence and add up the first three numbers to get the fourth number in the sequence. Continue this way to get the first 5 terms of the sequence.
For the next part of the question, we need to resort to geometric sequences. Find the nth term of the sequence and then equate each of the numbers to the nth term to find n.Â
{"text":"bold","word":"Definition"}
A Geometric sequence is a set of numbers where the ratio of any number to the previous number is a constant. Let us say the first number is "a" and the constant is "r". The nth term of a geometric sequence is given by {"math":"formula","eq":"ar^(^n^-^1^)"}
{"text":"bold","word":"Example"}
Consider this geometric sequence: 1, 2, 4, 8, 16,......
First term, a = 1, Common ratio r = 2
Second term = {"math":"formula","eq":"ar^(^2^-^1^)"} {"math":"formula","eq":"1 x 2^1=2"}
Third term = {"math":"formula","eq":"ar^(^3^-^1^)"}
{"math":"formula","eq":"1 x 2^2=4"}
{"text":"bold","word":"The Solution"}
{"text":"bold","word":"Part 1 - Finding the next 5 terms"}
Find the next five numbers in the sequence.
2 + 4 = 6
2 + 4 + 6 = 12
2 + 4 + 6 + 12 = 24
2 + 4 + 6 + 12 + 24 = 48
2 + 4 + 6 + 12 + 24 + 48 = 96
So the next five terms are {"text":"bold","word":"6, 12, 24, 48, 96."}
{"text":"bold","word":"Part 2"}
Find the nth term. The sequence starts at 6. So a = 6 and the common ratio, r = 2.
To check which of these numbers could not be numbers in the sequence, equate {"math":"formula","eq":"ar^(^n^-^1^)"} with each of the numbers. If n turns out to be a whole number, then the number is in the sequence, otherwise, it is not.
{"text":"bold","word":"(a) 49152"}
{"math":"formula","eq":"ar^(^n^-^1^) = 49152"}
{"math":"formula","eq":"6 x 2^(^n^-^1^) = 49152"}
Divide both sides by 6.
{"math":"formula","eq":"6 x 2^(^n^-^1^)/6= 49152/6"}
{"math":"formula","eq":"2^(^n^-^1^) = 8192"}
Prime factorize 8192.
{"math":"formula","eq":"2^(^n^-^1^) = 2^1^3"}
Equate the powers because the bases are the same.
n - 1 = 13
Add 1 on both sides.
n = 13 + 1 = 14
So 49152 is the 14th term in the sequence.
{"text":"bold","word":"(b) 64790"}
{"math":"formula","eq":"ar^(^n^-^1^) = 64790"}
{"math":"formula","eq":"6 x 2^(^n^-^1^) = 64790"}
Divide both sides by 6.
{"math":"formula","eq":"6 x 2^(^n^-^1^)/6= 64790/6"}
{"math":"formula","eq":"2^(^n^-^1^) = 10798.3"}
So "n" cannot be a whole number as the right-hand side cannot be written as 2 raised to some power, and 64790 cannot be a number in the sequence.
{"text":"bold","word":"(c) 24576"}
{"math":"formula","eq":"ar^(^n^-^1^) = 24576"}
{"math":"formula","eq":"6 x 2^(^n^-^1^) = 24576"}
Divide both sides by 6.
{"math":"formula","eq":"6 x 2^(^n^-^1^)/6= 24576/6"}
{"math":"formula","eq":"2^(^n^-^1^) = 4096"}
{"math":"formula","eq":"2^(^n^-^1^) = 2^1^2"}
Equate the powers because the bases are the same.
n - 1 = 12
Add 1 on both sides.
n = 12 + 1 = 13
So 24576Â is the 13th term in the sequence.
{"text":"bold","word":"(d) 12288"}
{"math":"formula","eq":"ar^(^n^-^1^) = 12288"}
{"math":"formula","eq":"6 x 2^(^n^-^1^) = 12288"}
Divide both sides by 6.
{"math":"formula","eq":"6 x 2^(^n^-^1^)/6= 12288/6"}
{"math":"formula","eq":"2^(^n^-^1^) = 2048"}
{"math":"formula","eq":"2^(^n^-^1^) = 2^1^1"}
Equate the powers because the bases are the same.
n - 1 = 11
Add 1 on both sides.
n = 11 + 1 = 12
So 12288Â is the 12th term in the sequence.
{"text":"bold","word":"(e) 34921"}
{"math":"formula","eq":"ar^(^n^-^1^) = 34921"}
{"math":"formula","eq":"6 x 2^(^n^-^1^) =Â 34921"}
Divide both sides by 6.
{"math":"formula","eq":"6 x 2^(^n^-^1^)/4=Â 34921/6"}
{"math":"formula","eq":"2^(^n^-^1^) = 5820.16"}
34921 is not a number in the sequence as the right-hand number cannot be written as 2 raised to some power.
{"text":"bold","word":"The numbers not in this sequence are:"}
64790 and 34921
{"text":"bold","word":"Learn about Geometric Sequences here"}
{"object":"video","url":"https://www.youtube.com/watch?v=pXo0bG4iAyg"}
