Help [ Maths-Sequence and Series problem]

[Mac g]

Determine the second term of an arithmetic progression. If the first term is 2 and if the first,third and seventh terms are in geometric progression.

[gmathintsdotcom]

Simple (just play around with the numbers)

 

First number is 2.

 

2, 3, 4, 5, 6, 7, 8 (this is an arithmetic progression/sequence)

 

First, Third, Seventh terms are 2, 4, 8 (this is geometric progression/sequence)

 

 

Hardcore math approach:

the first term of arithmetic: 2

second term: 2 + b (where b is the difference between each term)

third term: 2 + 2b

fourth term: 2 + 3b

....

seventh term: 2 + 6b

 

If the third term is in a geometric progression, it is k times the value of the first term

2k = 2 + 2b

 

If the seventh term is in a geometric progression, it is k times the value of the third term and k^2 the value of the first term.

2k^2 = 2 + 6b

 

2k = 2 + 2b

k = b + 1

 

Substitute this in the second equation

2(b+1)^2 = 2 + 6b

(b+1)^2 = 1 + 3b

b^2 + 2b + 1 = 1 + 3b

b^2 - b = 0

(b)(b-1) = 0

b = 0 or 1.

If b = 0, all the terms are the same

If b = 1, each term is 1 greater than the previous.

 

Since k = b + 1 (from above)

If b = 0, k = 1 all the terms are the same

If b = 1, k = 2 each term in the geometric sequence is twice the previous term.

 

So they are two possibilities,

If b=0: the sequence is 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, etc.

If b=1: the sequence is 2, 3, 4, 5, 6, 7, 8, 9, etc.

 

They probably want the latter choice.

Edited November 17, 2010 by gmathintsdotcom
Gratuitous link removed.