A-Level AQA Maths Pure Arithmetic Sequences & Series: Three consecutive terms in an ar

Three consecutive terms in an arithmetic sequence are $3e^{p}$, $5$, $3e^{r}$ - AQA - A-Level Maths Pure - Question 9 - 2017 - Paper 2

Question 9

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Three consecutive terms in an arithmetic sequence are $3e^{p}$, $5$, $3e^{r}$. Find the possible values of $p$. Give your answers in an exact form. Prove that ther... show full transcript

Worked Solution & Example Answer:Three consecutive terms in an arithmetic sequence are $3e^{p}$, $5$, $3e^{r}$ - AQA - A-Level Maths Pure - Question 9 - 2017 - Paper 2

Step 1

Find the possible values of p

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Answer

To determine the values of $p$ , we know that in an arithmetic sequence, the difference between consecutive terms is constant. Therefore, we equate the differences:

Set the first difference equal to the second: $3e^{p} - 5 = 3e^{r} - 5$
Simplifying this, we have: $3e^{p} - 5 = 3e^{r} - 5$
Thus, $3e^{p} = 3e^{r}$
Therefore, $e^{p} = e^{r}$ , which leads to $p = r$ .
Now, equate the first and last terms: $3e^{p} - 5 = 5 - 3e^{r}$
Rearranging gives: $3e^{p} + 3e^{r} = 10$ . Factoring out $3$ yields: $3(e^{p} + e^{r}) = 10$
Hence, $e^{p} + e^{r} = rac{10}{3}$ .
We have two expressions -- from the first part we established that $p = r$ . We can substitute: $2e^{p} = rac{10}{3}$
from which we find: $e^{p} = rac{5}{3}$
Then, taking the natural logarithm yields: $p = ext{ln} rac{5}{3}$ .

Step 2

Prove that there is no possible value of q

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Answer

To investigate whether $3e^{q}$ , $5$ , and $3e^{q}$ can be terms of a geometric sequence:

In a geometric sequence, the ratio of consecutive terms must be constant. Therefore: $rac{5}{3e^{q}} = rac{3e^{q}}{5}$ .
Cross-multiplying gives: $5 imes 5 = 3e^{q} imes 3e^{q}$
Simplifying, we have: $25 = 9(e^{q})^{2}$ .
Rearranging yields: $(e^{q})^{2} = rac{25}{9}$
which implies: $e^{q} = rac{5}{3} ext{ or } - rac{5}{3}$ .
Since $e^{q}$ is always positive, only $e^{q} = rac{5}{3}$ is valid. However, if we try to find the corresponding value of $q$ , we get: $q = ext{ln} rac{5}{3}$ which leads us back to the first sequence.
Thus, applying this value doesn't yield three distinct terms: we find that both endpoints equal $3e^{q}$ , contradicting the nature of a geometric sequence.

Therefore, we conclude that there is no possible value of $q$ for which these terms can be consecutive terms of a geometric sequence.

Three consecutive terms in an arithmetic sequence are $3e^{p}$, $5$, $3e^{r}$ - AQA - A-Level Maths Pure - Question 9 - 2017 - Paper 2

Worked Solution & Example Answer:Three consecutive terms in an arithmetic sequence are $3e^{p}$, $5$, $3e^{r}$ - AQA - A-Level Maths Pure - Question 9 - 2017 - Paper 2

Find the possible values of p

Prove that there is no possible value of q

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