SOLUTION: The sum of the 1st and 2nd terms of an AP is 4 and the 10th term is 19 Find d sum of the 5th and 6th terms

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Question 1098032: The sum of the 1st and 2nd terms of an AP is 4 and the 10th term is 19
Find d sum of the 5th and 6th terms

Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13198)   (Show Source): You can put this solution on YOUR website!

If the first term is a and the common difference is d, then the sum of the first two terms is a + (a+d) = a+2d, and the 10th term is a+9d.

[the sum of the first two terms is 4]
[the 10th term is 19]

That is a simple system of two linear equations. Solve them to find the values of a and d; then use a and d to find the values of the 5th and 6th terms.

Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
.
The MENTAL solution by the "guessing method"

 = 1;

 = 3;

d = 2 (the common difference);


Check:   =  + d*(10-1) = 1 + 2*9 = 19  ! Correct !


======================================================>


 = 1 + 4*2 = 9;

 = 1 + 5*2 = 11;

 +  = 9 + 11 = 20.

Solved.


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Surely, it can be solved algebraically

 +  = 4,      (1)
 =  = 19,   (2)

which gives you the system of two equations in two unknowns

 = 4,
 = 19,


You can solve it by any method you know/you want to get the same answer.


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There is a bunch of lessons on arithmetic progressions in this site:
    - Arithmetic progressions
    - The proofs of the formulas for arithmetic progressions
    - Problems on arithmetic progressions
    - Word problems on arithmetic progressions
    - Solved problems on arithmetic progressions


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

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