KOK YEN HAU's answer to Cans's Secondary 3 E Maths Singapore question.
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Hi I saw the answer has already been given in comment session I just want to help by giving you some explanations. Hope it helps you to understand better about the topic.
This topic is about number patterns. Very often, it involves a list of numbers that being called as “series” or “progression”, and you would require to observe the patterns in relation to the changes of the numbers, sort of figuring out the logic of determining the next number. The numbers/elements in the series is often being called as “term”.
Depending on what kind of question that you are dealing with, sometimes you are given some terms directly in a series as the start, sometimes it is indirect, such as your question here where you need to manually draw and count to get the value first few terms, then observe the logic/pattern from there onwards.
The series in your question: 1, 5, 9, 13, … is one of the most basic one called arithmetic series/ arithmetic progression, where two consecutive terms always have same difference, usually being called as “common difference”, the difference can be positive or negative. In your question, the common difference of the series, says d= +4. Along with the value of first term, says a, in your question, a=1, both properties are sufficient to fully define the series, in another word, once knowing the first term and common difference, you can derive any term in the series from there.
Some basic formulas of arithmetic progression include the value of n-th term = a+(n-1)d, sum of first n-th term = (a + n-th term)(n)/2
From your question, a hexagonal number is the sum of first n-th term in the arithmetic progression 1, 5, 9, 13, … (with a=1, d=4)
Which means T1=sum of first term, T2=sum of first 2 terms, and so on…
By substituting a=1 and d=4 into the formula above, you can get Tn=n(2n-1)
A good habit is to verify your equation T1= 1*(2-1) = 1 which is correct, T2 = 2*(4-1) = 6 which is same as the 1+5 in the question, so you can be sure now it is correct.
Then you can just set n=1,2,3,4,5,6,7 to get all the hexagonal number T1, T2, T3, T4,… and see if 91 shows up among them, for your question part (c)..
This topic is about number patterns. Very often, it involves a list of numbers that being called as “series” or “progression”, and you would require to observe the patterns in relation to the changes of the numbers, sort of figuring out the logic of determining the next number. The numbers/elements in the series is often being called as “term”.
Depending on what kind of question that you are dealing with, sometimes you are given some terms directly in a series as the start, sometimes it is indirect, such as your question here where you need to manually draw and count to get the value first few terms, then observe the logic/pattern from there onwards.
The series in your question: 1, 5, 9, 13, … is one of the most basic one called arithmetic series/ arithmetic progression, where two consecutive terms always have same difference, usually being called as “common difference”, the difference can be positive or negative. In your question, the common difference of the series, says d= +4. Along with the value of first term, says a, in your question, a=1, both properties are sufficient to fully define the series, in another word, once knowing the first term and common difference, you can derive any term in the series from there.
Some basic formulas of arithmetic progression include the value of n-th term = a+(n-1)d, sum of first n-th term = (a + n-th term)(n)/2
From your question, a hexagonal number is the sum of first n-th term in the arithmetic progression 1, 5, 9, 13, … (with a=1, d=4)
Which means T1=sum of first term, T2=sum of first 2 terms, and so on…
By substituting a=1 and d=4 into the formula above, you can get Tn=n(2n-1)
A good habit is to verify your equation T1= 1*(2-1) = 1 which is correct, T2 = 2*(4-1) = 6 which is same as the 1+5 in the question, so you can be sure now it is correct.
Then you can just set n=1,2,3,4,5,6,7 to get all the hexagonal number T1, T2, T3, T4,… and see if 91 shows up among them, for your question part (c)..
Date Posted:
1 year ago
Ahh thank you so much! I understand much better now. Thank you for taking the time to explain to me, I really appreciate it.
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