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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE General & Further Mathematics => Topic started by: Yendall on September 24, 2012, 06:41:55 pm

Title: Finding a general term when given the sum?
Post by: Yendall on September 24, 2012, 06:41:55 pm: Okay, so this type of question constantly gets me everytime I attempt a practice exam:
This is from NEAP 2010 trial examination one:

Question 6
A certain arithmetic sequence has first term 27. The sum of the first 5 terms is 25.
The second term is
A. -11
B. 11
C. 16
D. 26
E. 38

Would you solve via arithmetic?
We know that $S_5 = 25$ and $T_1 = 27$

Would we use $S_n = \frac{n}{2} (a + t_n)$ ?

So subbing in $S_5 = 25$ and $T_1 = 27$ to find the value of the fifth term:
$25 = \frac{5}{2} (27 + t_5)$

$50 = 5(27 + t_5)$

$10 = 27 + t_5$

$-17 = t_5$

Then finding $d$ from two values $t_1 = 27$ and $t_5 = -17$

$d = \frac{t_b - t_a}{b - a}$

$d = \frac{t_5 - t_1}{5 - 1}$

$d = \frac{-17 - 27}{4}$

$d = \frac{-44}{4}$

$d = -11$

Then finding $t_2$ using $d = -11$

$27 + -11 = 16$

Therefore the answer will be C. 16

Is this the correct way of solving this type of question?
Title: Re: Finding a general term when giving the sum
Post by: paulsterio on September 24, 2012, 06:47:24 pm: Quote from: Yendall on September 24, 2012, 06:41:55 pm
Question 6
A certain arithmetic sequence has first term 27. The sum of the first 5 terms is 25.
The second term is
A. -11
B. 11
C. 16
D. 26
E. 38

Do it by first principles,

So you know that the first term is 27 and the sum of the first five terms is 25.

What you want to do now is:

$27 \times 5 + x + 2x + 3x + 4x = 25$

Solve for $x$

$x = -11$

Common difference is -11 and hence, the second term will be 16 (C).

Alternatively, use the formula

$S_n=\frac{n}{2}(2a + (n-1)d)$

$25 = \frac{5}{2}(2 \times 27 + 4d)$

Solve for d and you should be able to evaluate from there.
Title: Re: Finding a general term when giving the sum
Post by: Yendall on September 24, 2012, 06:49:29 pm: Quote
Do it by first principles,

So you know that the first term is 27 and the sum of the first five terms is 25.

What you want to do now is:

$27 \times 5 + x + 2x + 3x + 4x = 25$

Solve for $x$

$x = -11$

Common difference is -11 and hence, the second term will be 16 (C).

Alternatively, use the formula

$S_n=\frac{n}{2}(2a + (n-1)d)$

$25 = \frac{5}{2}(2 \times 27 + 4d)$

Solve for d and you should be able to evaluate from there.
Oh thank you paul! i never thought of simply solving for d with that equation. Would you recommend that method over solving for x?
Title: Re: Finding a general term when given the sum?
Post by: paulsterio on September 24, 2012, 06:54:06 pm: If you wanted to be extremely cute though, you could technically do it this way:

Let $x$ be the second term:

$27 + x + 2x - 27 + 3x - 27*2 + 4x - 27*3 = 25$

$x = 16$

That's probably what I would have done.

But honestly though, it doesn't really matter, any way that gets the answer is fine, it just depends on how you understand it.

Of course, the formula is the SAFEST way, but you have to know when to apply the right formula, using first principles is also good because it always works.
Title: Re: Finding a general term when given the sum?
Post by: Yendall on September 24, 2012, 06:57:54 pm: Quote from: paulsterio on September 24, 2012, 06:54:06 pm
If you wanted to be extremely cute though, you could technically do it this way:

Let $x$ be the second term:

$27 + x + 2x - 27 + 3x - 27*2 + 4x - 27*3 = 25$

$x = 16$

That's probably what I would have done.

But honestly though, it doesn't really matter, any way that gets the answer is fine, it just depends on how you understand it.

Of course, the formula is the SAFEST way, but you have to know when to apply the right formula, using first principles is also good because it always works.
Okay awesome, I think i'll work it out using first principles if i'm uncertain as to which equation fits the question. I think figuring both out would waste a lot of time in the exam.
Title: Re: Finding a general term when given the sum?
Post by: paulsterio on September 24, 2012, 07:01:22 pm: Yeah, obviously doing both ways would be a waste of time - first principles is always a good way to go about things.

In this case, if you let x be the common difference, you can see that the first term will be 27, then the next term will be 27 + x and then the term after that will be 27 + 2x...etc.

So essentially what you have is 27 + 27 + x + 27 + 2x + 27 + 3x + 27 + 4x (which are the first 5 terms) - simplify it to 5*27 + 10x and let it equal to 25, which is the sum and solve for x and you have the difference, then just figure out the 2nd term!
Title: Re: Finding a general term when given the sum?
Post by: Yendall on September 24, 2012, 07:14:22 pm: Quote from: paulsterio on September 24, 2012, 07:01:22 pm
Yeah, obviously doing both ways would be a waste of time - first principles is always a good way to go about things.

In this case, if you let x be the common difference, you can see that the first term will be 27, then the next term will be 27 + x and then the term after that will be 27 + 2x...etc.

So essentially what you have is 27 + 27 + x + 27 + 2x + 27 + 3x + 27 + 4x (which are the first 5 terms) - simplify it to 5*27 + 10x and let it equal to 25, which is the sum and solve for x and you have the difference, then just figure out the 2nd term!
And you can apply this to basically anything? Say I wanted to find $d$ and $t_2$ when $t_1 = 10$ and $S_6 = 30$

$10 + 10 + x + 10 + 2x + 10 + 3x + 10 + 4x + 10 + 5x = S_6$

$6*10 + 15x$

$solve(30 = 6*10 + 15x, x) = -2$

Therefore,
$10 + -2 = t_2 = 8$
Title: Re: Finding a general term when given the sum?
Post by: paulsterio on September 24, 2012, 07:19:08 pm: Yep, that's correct :)
Title: Re: Finding a general term when given the sum?
Post by: Yendall on September 24, 2012, 07:24:34 pm: Quote from: paulsterio on September 24, 2012, 07:19:08 pm
Yep, that's correct :)
Awesome, thank you :)