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Aritmetic Sequence

by Sylvia Andrew
(Kota Kinabalu)











































given the third term of an arithmetic sequence less than the fourth term by three. The seventh term is two times the firth term. Find the common different and the first term

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Jul 21, 2011
Arithmetic Sequence
by: Staff


The question:

given the third term of an arithmetic sequence less than the fourth term by three. The seventh term is two times the firth term. Find the common difference and the first term



The answer:


An arithmetic sequence has the form:

a_n = a_1 + (n - 1) * (d)


a_n = a with a subscript of n (this is the nth term in the series)

a_1 = a with a subscript of 1 (this is the 1st term in the series)

n = number of terms

d = difference between consecutive terms


Applying the notation to your problem

the third term of an arithmetic sequence less than the fourth term by three

d (the common difference) = 3


The seventh term is two times the firth term.

a_7 = 2 * a_5



Find the first term

a_n = a_1 + (n - 1) * (d)

a_7 = a_1 + (7 - 1) * (3)

a_7 = a_1 + (6) * (3)

a_7 = a_1 + 18


a_5 = a_1 + (5 - 1) * (3)

a_5 = a_1 + (4) * (3)

a_5 = a_1 + 12




a_7 = 2 * a_5

Therefore,

a_7 = 2 * (a_1 + 12)


we now have two equations with two unknowns


a_7 = a_1 + 18

a_7 = 2 * (a_1 + 12)


simplify the 2nd equation

a_7 = 2 * (a_1 + 12)

a_7 = 2*a_1 + 2*12

a_7 = 2a_1 + 24


solve for the first term a_1 using the addition/subtraction method

a_7 = 2a_1 + 24

a_7 = a_1 + 18



a_7 = 2a_1 + 24

-(a_7 = a_1 + 18)
-----------------------

a_7 - a_7 = 2a_1 - a_1 + 24 - 18

0 = 2a_1 - a_1 + 24 - 18

0 = a_1 + 24 - 18

0 = a_1 + 6

0 - 6 = a_1 + 6 - 6

- 6 = a_1 + 6 - 6

- 6 = a_1 + 0

- 6 = a_1

a_1 = - 6


the final answer is:

a_1 (the first term in the sequence) = - 6
d (the common difference) = 3




check the answer: Is a_7 = 2 * a_5 valid for the solutions we have calculated?


Calculate a_7

a_n = a_1 + (n - 1) * (d)

a_7 = a_1 + (7 - 1) * (3)

a_7 = -6 + (7 - 1) * (3)

a_7 = -6 + 18

a_7 = 12


Calculate a_5

a_5 = a_1 + (5 - 1) * (3)

a_5 = -6 + (4) * (3)

a_5 = -6 + 12

a_5 = 6


a_7 = 2 * a_5

12 = 2 * 6

12 = 12, OK (a_7 does = 2 * a_5)




Thanks for writing.

Staff
www.solving-math-problems.com



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