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A level maths

.Help.me.

I need help in answering this question, I don't even know where to start.
Working out would be appreciated

The positive integers x, y and z are the first, second and third terms, respectively, of an arithmetic progression with common difference -4.
Also, x, 15/y and z are the first, second and third terms, respectively, of a geometric progression.
(a) Show that y satisfies the equation y^4-16y²-225=0,
(b) Hence determine the sum to infinity of the geometric progression.

Reply 1

MindMax2000

Universities Forum Helper

Original post by .Help.me.

This is going to be difficult to advise due to the fact we are not allowed to offer answers.

I would start with the following:

•

Formula for arithmetic progression is tn = a + (n-1)d
You are given the first 3 terms of the arithmetic progression, the first term x, and the common difference of -4.

•

You want to only have y in the equation, so you need to isolate and eliminate x and z from the equations. Find Term 3.

•

As the question in part b implies that the geometric progression is somehow related to the arithmetic progression, there must be common terms.

•

As you would later notice the third term in the geometric progression is z, which is the same as the third term in the arithmetic progression.

•

The formula for the geometric progresion is tn = ar^(n-1)
Find the second and third terms fo the geometric sequence.

•

You should be able to notice that the third terms for both the geometric and arithmetic sequences can be eliminated via simultaneous equations.

•

Isolate x in the second terms in both the geometric and arithmetic sequences to eliminate x (if you need to)

•

Notice you need to find r in the geometric sequence in order to find the sum to infinity.

•

It would really help if you note down all the relevant formulas in your answer before you start anything; it would save you from thinking as hard.

Reply 2

.Help.me.

Original post by MindMax2000

This is going to be difficult to advise due to the fact we are not allowed to offer answers.

I would start with the following:

•

Formula for arithmetic progression is tn = a + (n-1)d
You are given the first 3 terms of the arithmetic progression, the first term x, and the common difference of -4.

•

You want to only have y in the equation, so you need to isolate and eliminate x and z from the equations. Find Term 3.

•

As the question in part b implies that the geometric progression is somehow related to the arithmetic progression, there must be common terms.

•

As you would later notice the third term in the geometric progression is z, which is the same as the third term in the arithmetic progression.

•

The formula for the geometric progresion is tn = ar^(n-1)
Find the second and third terms fo the geometric sequence.

•

You should be able to notice that the third terms for both the geometric and arithmetic sequences can be eliminated via simultaneous equations.

•

Isolate x in the second terms in both the geometric and arithmetic sequences to eliminate x (if you need to)

•

Notice you need to find r in the geometric sequence in order to find the sum to infinity.

•

It would really help if you note down all the relevant formulas in your answer before you start anything; it would save you from thinking as hard.

For part a would I need to just turn it into a quadratic and solve for y? And would that y be used in part b?

Reply 3

mqb2766

Original post by .Help.me.

For part a would I need to just turn it into a quadratic and solve for y? And would that y be used in part b?

From the arithmetic sequence you can easily get expressions for both x and z in terms of y, then use these in the geometric sequence where
z = ar^2
where a is .. and r can be obtained from the 1st and 2nd terms in the sequence. Obviously it all needs to be expressed in terms of y.

(edited 11 months ago)

Reply 4

MindMax2000

Universities Forum Helper

Original post by .Help.me.

For part a would I need to just turn it into a quadratic and solve for y? And would that y be used in part b?

Nope.

For part a), the question is about showing how you arrive at the equation y^4-16y²-225=0, There is not mention of having to solve it here yet.
For part b), you are asked to find the sum to infinity of the geometric progression, which should require you to solve the quadratic for y, then find what x and z are