Question 1107227
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<pre>
For each arithmetic progression,  the sum of terms equally remoted from the first and the last terms is the constant value.


In other words, if  {{{a[1]}}}, {{{a[2]}}}, {{{a[3]}}}, . . . , {{{a[n-1]}}}, {{{a[n]}}} is an arifmetic progression of n terms, then

    {{{a[1]}}} + {{{a[n]}}} = {{{a[2]}}} + {{{a[n-1]}}} = {{{a[3]}}} + {{{a[n-2]}}}  and so on.


For the AP of 10 terms

   {{{a[1]}}} + {{{a[10]}}} = {{{a[2]}}} + {{{a[9]}}} = {{{a[3]}}} + {{{a[8]}}} = . . . = {{{a[5]}}} + {{{a[6]}}}.


Hence, the sum  {{{a[1]}}} + {{{a[2]}}} + . . . + {{{a[10]}}}  is 5 times taken the sum {{{a[5]}}} + {{{a[6]}}}:

    {{{a[1]}}} + {{{a[2]}}} + . . . + {{{a[10]}}} = {{{5*(a[5] + a[6])}}}.   (1)


        This formula is PARALLEL (or similar) to the general formula for the sum of an AP:    {{{S[n]}}} = {{{((a[1]+a[n])/2)*(n/2)}}}.


From the formula (1) we can find the sum  {{{a[5]+a[6]}}}.  It is

    {{{a[5]+a[6]}}} = {{{S[10]/5}}} = {{{175/5}}} = 35.


We also know that {{{a[5]}}} = 16.  

It gives  {{{a[6]}}} = 35 - 16 = 19.


Thus two neighbor terms of the AP are 16 and 19; they differ by 3 = 19-16, which is the common difference d of the progression.


Having {{{a[5]}}} = 16 and d = 3,  we can find  

{{{a[1]}}} = {{{a[5]-4*d}}} = 16 - 4*3 = 4,

{{{a[25]}}} = {{{a[1]+24*d}}} = 4 + 24*3 = 76.


Now, to find  {{{S[25]}}}, apply the general formula

{{{S[25]}}} = {{{((a[1]+a[25])/2)*25}}} = {{{((4+76)/2)*25}}} = 1000.


<U>Answer</U>.  {{{S[25]}}} = 1000.
</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.