Question 257916
sum of the terms of an arithmetic series is given by the formula:


s[n] = n * (a[1] + a[n])/2


s[n] = sum of the n terms of the sequence.
a[1] = first term in the sequence.
a[n] = nth term in the sequence.
n = number of terms in the sequence.


the formuula for the nth term in the sequence is:


a[n] = a[1] + (n-1) * d


d is the common difference.


you will use both these formulas to find the answer.


you are given:


sum of the first 9 terms in a sequence is 162
sum of the first 12 terms in the same sequence is 288.


the formnula for the sum of the first n terms in a sequence is:


s[n] = n * (a[1] + a[n])/2


substitute for the first 9 terms to get:


162 = 9 * (a[1] + a[9])/2 (first equation)


substitute for the first 12 terms to get:


288 = 12 * (a[1] + a[12])/2 (second equation)


multiply both sides of the first equation by 2 and divide both sides of the first equation by 9 to get:


a[1] + a[9] = 36 (third equation)


multiply both sides of the second equation by 2 and divide both sides of the second equation by 12 to get:


a[1] + a[12] = 48 (fourth equation)


subtract third equation from fourth equation to get:


a[12] - a[9] = 12


since 12 - 9 = 3, there are 3 terms between a[9] and a[12].


12/3 = 4 which means the common difference between each term is 4.


the nth term in each sequence is given by the formula:


a[n] = a[1] + (n-1) * d


we now know that d = 4


we solve for a[9] to get:


a[9] = a[1] + 8*4 = a[1] + 32


we take our first equation and plug in our new found values.


first equation is:


162 = 9 * (a[1] + a[9])/2 (first equation)


substitute a[1] + 32 for a[9] to get:


162 = 9 * (a[1] + a[1] + 32)/2


combine like terms to get:


162 = 9 * (2* a[1] + 32)/2


multiply both sides of this equation by 2 and divide both sides of this equation by 9 to get:


36 = 2 * a[1] + 32


subtract 32 from both sides of this equation to get:


36-32 = 2 * a[1] which becomes:


4 = 2 * a[1].


divide both sides of this equation by 2 to get:


a[1] = 2.


we now know a[1] and we know d so we can construct the sequence.


formula for nth term in the sequence is:


a[n] = a[1] + (n-1) * d


since d = 4,...


the 9th term in the sequence is:


a[9] = 2 + 8*4 = 2 + 32 = 34


the 12th term in the sequence is:


a[12] = 2 + 11*4 = 2 + 44 = 46


the sequence is:

<pre>         
term:         <font color = red>1</font>  2  3   4   5   6   7   8   <font color = red>9</font>   10  11  <font color = red>12</font>

sequence:     <font color = red>2</font>  6  10  14  18  22  26  30  <font color = red>34</font>  38  42  <font color = red>46</font>
</pre>
2 is the 1st term.
34 is the 9th term.
46 is the 12th term.


common difference is 4.