Question 1208027
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Find the {{{highlight(cross(valee))}}} <U>value</U> of k and the {{{highlight(cross(17))}}} <U>17-th</U> term of each of the following arithmetic sequence
6) 2k+1,5k-3, 7k-2
7) 7k+2,5k+4,4k-5
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I will solve first part, &nbsp;leaving the second part to you as an exercise, 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;because the second part is a &nbsp;TWIN &nbsp;to the first part.



<pre>
Since three terms {{{a[1]}}},  {{{a[2]}}} and {{{a[3]}}} form an arithmetic progression,
the difference  {{{a[2]-a[1]}}} is the same as the difference  {{{a[3]-a[2]}}}
(as each this difference is simply d, the common difference of the AP).


So, we write

    (5k-3) - (2k+1) = (7k-2) - (5k-3).


From this equation, we find the value of k

    5k - 3 - 2k - 1 = 7k - 2 - 5k + 3,

    3k - 4 = 2k +1

    3k - 2k = 1 + 4

        k   =   5.


Now we know the value of k and can restore the values of the first three twrms of this AP

    {{{a[1]}}} = 2*5+1 = 11;

    {{{a[2]}}} = 5*5-3 = 22;

    {{{a[3]}}} = 7*5-2 = 33.


Thus, the progression has first term 11 and the common difference d = 22-11 = 11.


At this point, we know everything about this AP and can easily find each of its terms.


In particular, the 17-th term of this AP is  {{{a[17]}}} = {{{a[1]}}} + d*(n-1) = 11 + 11*(17-1) = 11 + 11*16 = 187.    <U>ANSWER</U>
</pre>

Solved.


This solution is your TEMPLATE to solve many other similar problems.


In particular, the other part is a TWIN and can be solved by the same way.


Literally follow my solution, step by step.


You may report me about your progress.