Question 1121963
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<pre>
Use the formula for the sum of the first n terms of an arithmetic progression


{{{S[n]}}} = {{{a[1] + a[2] + a[3] + ellipsis + a[n]}}} = {{{(a[1] + (n-1)*d/2)*n}}}.


Since an arithmetic sequence a1,a2,......a100 has the sum of 15000, you have first equation


{{{(a[1] + (99/2)*d)*100}}} = 15000,   or


{{{2a[1] + 99d}}} = 300.



The second progression is also an arithmetic progression with the first term of {{{a[1]+2d}}}  and the common difference of 3d.


Therefore, applying the same formula for the second progression (which has 33 terms), you get


{{{(a[1]+2d + (32/2)*(3d))*33}}} = 5016,   or


{{{a[1] + 50d}}} = 152.


Thus you have this system of 2 equations in 2 unknowns


{{{2a[1] + 99d}}} = 300.    (1)

{{{a[1] + 50d}}} = 152.     (2)


Multiply eq(2) by 2 (both sides) and then subtract the eq(1) from it. You will get


d = 304 - 300 = 4.


Then from (2)  {{{a[1]}}} = 152 - 50*4 = 152 - 200 = - 48.


<U>Answer</U>.   {{{a[1]}}} = -48;  d = 4.
</pre>

Solved.


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There is a bunch of lessons on arithmetic progressions in this site:

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Arithmetic-progressions.lesson>Arithmetic progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/The-proofs-of-the-formulas-for-arithmetic-progressions.lesson>The proofs of the formulas for arithmetic progressions</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Problems-on-arithmetic-progressions.lesson>Problems on arithmetic progressions</A>  

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Word-problems-on-arithmetic-progressions.lesson>Word problems on arithmetic progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Mathematical-induction-and-arithmetic-progressions.lesson>Mathematical induction and arithmetic progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/One-characteristic-property-of-arithmetic-progressions.lesson>One characteristic property of arithmetic progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Solved-problems-on-arithmetic-progressions.lesson>Solved problems on arithmetic progressions</A> 



Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic <U>"Arithmetic progressions"</U>.



Save the link to this textbook together with its description


Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson


into your archive and use when it is needed.