Farhanthestudent005
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 HSC
 2022
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Use that to our advantage we can say that we can go with first term as 4 and then second term as 8 and etc. If you have mastery of 4 times tables you will notice that the 25th term is 100 but for this question it is instead 99 so therefore this will be the 26th term because we are using 4n1 to our advantage here by first using the 4 times tables where we know that 4 times 26 is 104 but take away 1 gives 103 so thereby we will have the 26th year.Here are some hints a which is the start is 3 and the common difference is 4.
Nice way of working however I would go with 4+8+12+...+76+80 using the fact that the series uses a familiar pattern that the user knows of and then use the way how Gauss worked on this question and then take away 20 for the 20 terms.3, 7, 11, 15... etc form an arithmetic sequence.
Are you familiar with the formula for calculating the sum of the first n terms of an arithmetic sequence? Do you understand how the formula was derived?
S = n [2a + (n1)d] /2
S is the sum of the first n terms
n is the number of terms
a is the initial term
d is the difference between terms
For part b) of the question, we are asked to find n and are given:
a = 3
d = 4
S > 100
Substituting these values:
n [6 + (n1)4] /2 > 100
Simplify the left side:
n (2n + 1) > 100
Substitute the inequality and solve the equivalent quadratic equation: 2n^2 + n  100 = 0
n = (1 + sq rt( 1 + 800)) / 4 ~ 6.8
Therefore, the next higher rounded integer n=7 is the solution to the inequality, so the answer is in the seventh year.
For part c) of the question, we are asked to find S and are given:
a = 3
d = 4
n = 20
Substituting these values:
S = 20 [6 + (19)4] /2
= 820
I did appreciate your approach of trying to teach the concepts which I think will benefit other viewers, but I am mindful that this OP is sitting HSC this year, so I thought combining the YouTube video and showing how to directly apply the equation would be the most efficient reply for him. I did wonder whether this was the correct maths section on BoS to pose a question on arithmetic series as I thought this was covered in Yr 11 rather than in extn 1, though I’m not all that familiar with the curriculum details.Nice way of working however I would go with 4+8+12+...+76+80 using the fact that the series uses a familiar pattern that the user knows of and then use the way how Gauss worked on this question and then take away 20 for the 20 terms.
The arithmetic series @Eagle Mum is taught very early in Year 12 Maths Advanced. I mean if the class is very quick at learning it can be taught around Year 11 term 3. This is a very easy concept so that is why it is put in early.I did appreciate your approach of trying to teach the concepts which I think will benefit other viewers, but I am mindful that this OP is sitting HSC this year, so I thought combining the YouTube video and showing how to directly apply the equation would be the most efficient reply for him. I did wonder whether this was the correct maths section on BoS to pose a question on arithmetic series as I thought this was covered in Yr 11 rather than in extn 1, though I’m not all that familiar with the curriculum details.