MHB Arithmetic and geometric sequence

AI Thread Summary
The discussion focuses on solving problems related to arithmetic and geometric sequences. For the arithmetic sequence, the 1st term is calculated as 58/5, and the 5th term is -98/5, with no term equating to -70. In the geometric sequence, the 3rd and 5th terms are derived from two potential sequences, yielding values of 8 and 32 for one sequence, and -8 and -32 for the other. The calculations involve determining common differences and ratios based on given terms. This analysis provides a clear understanding of the sequences in question.
paix1988
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*I am struggling with arithmetic and geometric sequences.

if the 4th term is m-8, 6th term 8m+3 and 8th term is 10m-5
Calculate the 1st and 5th term
Which term will have a value of -70The 4th term of geometric sequence is -16 and the 6th term is -64. Calculate the 3rd and 5th terms.

thank you for your assistance
 
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Hello and welcome to MHB! paix1988 :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?
 
paix1988 said:
*I am struggling with arithmetic and geometric sequences.

if the 4th term is m-8, 6th term 8m+3 and 8th term is 10m-5
Calculate the 1st and 5th term
Which term will have a value of -70The 4th term of geometric sequence is -16 and the 6th term is -64. Calculate the 3rd and 5th terms.

thank you for your assistance

For the benefit of the community, I am going to post solutions.

1.) We are given the following information regarding an arithmetic progression:

$$a_4=m-8,\,a_6=8m+3,\,a_8=10m-5$$

If we denote the common difference as $d$, then from this we know:

$$2d=(8m+3)-(m-8)=7m+11$$

$$2d=(10m-5)-(8m+3)=2m-8$$

Equating the two results, we obtain:

$$7m+11=2m-8\implies m=-\frac{19}{5}\implies d=-\frac{39}{5}$$

Now, in general, we have:

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

And we know:

$$a_4=a_1+3d$$

$$m-8=a_1+3d$$

$$-\frac{59}{5}=a_1-\frac{117}{5}$$

$$a_1=\frac{58}{5}$$

Hence, the general term is:

$$a_n=\frac{58}{5}+(n-1)\left(-\frac{39}{5}\right)=\frac{58-39(n-1)}{5}$$

Thus, the 5th term is:

$$a_5=\frac{58-39(5-1)}{5}=-\frac{98}{5}$$

To find which term has a value of -70, we may write:

$$\frac{58-39(n-1)}{5}=-70$$

$$58-39(n-1)=-350$$

$$-39(n-1)=-408$$

$$n-1=\frac{136}{13}$$

We will not get an integral value for $n$, thus none of the terms of this AP has a value of -70.

2.)The $n$th term of a GP is:

$$a_n=ar^n$$

We know that:

$$\frac{g_6}{g_4}=\frac{-64}{-16}=4=r^2\implies r=\pm2\implies a=-1$$

Thus, there are two possible GPs satisfying the given information:

a) $$a_n=-(-2)^n\implies a_3=8,\,a_5=32$$

b) $$a_n=-2^n\implies a_3=-8,\,a_5=-32$$
 
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