SOLUTION: Question: "The sum of 40 terms of a certain arithmetic sequence is 430, while the sum of 60 terms is 945. Determine the nth term of the arithmetic sequence." How do I approach

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Question 1163799: Question: "The sum of 40 terms of a certain arithmetic sequence is 430, while the sum of 60 terms is 945. Determine the nth term of the arithmetic sequence."
How do I approach this problem? I went through the questions that are already answered on this site as well as its Algebra II text book. I am still at a loss.
Thanks
Found 3 solutions by greenestamps, ikleyn, MathTherapy:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


If the reference you have is like most, it tells you the sum of n terms of an arithmetic sequence is



where n is the number of terms and a(1) and a(n) are the first and last terms.

While that formula is valid, it is not intuitively obvious. Why should you divide the number of terms by 2 and then multiply by the sum of the first and last terms?

I VERY MUCH prefer this alternative form of the formula:



In that form, the formula says the sum is the average of the terms, multiplied by the number of terms. That makes perfect sense -- that is what average means.

So let's use that formula with the two pieces of given information: the sum of the first 40 terms is 430, and the sum of the first 60 terms is 945.

Use a as the first term and d as the common difference.

Sum of first 40 terms is 430....

number of terms: n = 40
first term: a
40th term: a+39d
sum: [1]

Sum of first 60 terms is 945....

number of terms: n = 60
first term: a
60th term: a+59d
sum: [2]

Now solve [1] and [2] simultaneously.


[3]


[4]

Comparing [3] and [4] gives us




Substituting in [3]...





The first term of the sequence is a=1; the common difference is d=1/2.

The formula for the n-th term of the sequence is




CHECK:
40 terms....
1st term: 1
40th term: 1+39/2 = 41/2
sum: correct

60 terms....
1st term: 1
60th term: 1+59/2 = 61/2
sum: correct
Answer by ikleyn(52788)   (Show Source): You can put this solution on YOUR website!
.
The sum of 40 terms of a certain arithmetic sequence is 430, while the sum of 60 terms is 945.
Determine the nth term of the arithmetic sequence.
~~~~~~~~~~~~~~ 


The sum of any AP is the average of its extreme terms times the number of terms.

So, from the condition


     = 430.

     = 945.



It gives

     =  = 21.5     (1)

     =  = 31.5     (2)



Now subtract equation (1) from equation (2)

     -  = 31.5 - 21.5 = 10.



But    -  = 20d,  where "d"  is the common difference.

Hence,  d = 10/20 = 0.5.



Next,  from equation (1)

     +  = 21.5;

hence,

     =  = 1.



Now the n-th term is    =  + (n-1)*d = 1 + (n-1)*0.5.        ANSWER

Solved.

-----------------

My lessons on arithmetic progressions in this site are
    - Arithmetic progressions
    - The proofs of the formulas for arithmetic progressions
    - Problems on arithmetic progressions
    - Word problems on arithmetic progressions
    - Chocolate bars and arithmetic progressions
    - One characteristic property of arithmetic progressions
    - Solved problems on arithmetic progressions
    - Calculating partial sums of arithmetic progressions
    - Finding number of terms of an arithmetic progression
    - Advanced problems on arithmetic progressions
    - Interior angles of a polygon and Arithmetic progression
    - Problems on arithmetic progressions solved MENTALLY

Also, you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

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


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.



Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

Question: "The sum of 40 terms of a certain arithmetic sequence is 430, while the sum of 60 terms is 945. Determine the nth term of the arithmetic sequence."
How do I approach this problem? I went through the questions that are already answered on this site as well as its Algebra II text book. I am still at a loss.
Thanks
Use the following formula for the SUM of an A.P.: 
                                                  
                                                   ------ Multiplying by LCD, 2 ---- eq (i)
 
                                                  
                                                   ------ Multiplying by LCD, 2 ---- eq (ii) 

                                                  20 = 40d ------- Subtracting eq (i) from eq (ii)
                                                  Common difference, or 

                                                   ------- Substituting ½ for d in eq (i)
                                                  
                                                  

Use the following formula for the nth term of an A.P.:

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