Question 1136303
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<pre>
(i)   "the sum of the fourth and ninth terms is 58"  means

      {{{a[4] + a[9]}}} = 58,    or

      {{{a[1] + 3d + a[1]+8d}}} = 58,   or

      {{{2a[1] + 11d}}} = 58.     (1)



(ii)  "the sum of the first 26 terms is 390"  means  that

      {{{(a[1] + a[26])/2}}} = {{{390/26}}} = 15,

      since the sum of the first n terms of any AP is the average of the first and n-th terms taken n times.


      The last equation means that

      {{{a[1] + a[1] + 25d}}} = 30,   or

      {{{2a[1] + 25d}}} = 30.      (2)



Now subtract eq(1) from eq(2).  You will get


       25d - 11d = 30 - 58,   or

       14d       = -28

which implies  d = -2.


Then from eq(1) you have  {{{a[1]}}} = 58 - 11*(-2) = 58 + 22 = 80.


Thus the AP has  {{{a[1]}}} = 80  and  d= -2.    <U>ANSWER to question (a)</U>


Now it is obvious that the first 39 terms of the AP are uniformly decreasing from 80 to 2 with the step -2;

the 40-th term is 0 (zero);

next 39 terms from  {{{a[41]}}}  to  {{{a[79]}}}  are negative from -2 to -80 and the sum of the first 79 terms is equal to zero.


The smallest integer value of "n" for which the sum to n terms is negative is 80.    <U>ANSWER to question (b)</U>.
</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/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> 

&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/Mathematical-induction-for-sequences-other-than-arithmetic-or-geometric.lesson>Mathematical induction for sequences other than arithmetic or geometric</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.