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9 (a) 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
9 (a) 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. 9 (b) Pro... show full transcript
Worked Solution & Example Answer:9 (a) 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 $p$
Answer
To find the possible values of , we start by setting up the condition for the terms to be in an arithmetic sequence. The difference between the first and second term should equal the difference between the second and third term:
This simplifies to:
Dividing both sides by 3 gives:
Taking the natural logarithm of both sides results in:
Next, we also set the difference condition as:
Rearranging gives:
Substituting with , we have:
or:
Dividing both sides by 6 yields:
e^{p} = rac{10}{6} = rac{5}{3}
Taking the natural logarithm of both sides gives:
p = ext{ln} rac{5}{3}
Thus, the possible values of in exact form is:
p = ext{ln} rac{5}{3}.
Step 2
Prove that there is no possible value of $q$ for which $3e^{q}$, $5$, $3e^{r}$ are consecutive terms of a geometric sequence.
Answer
To prove this, we assume that , , and are consecutive terms of a geometric sequence, which means that the ratio between consecutive terms must be equal:
rac{5}{3e^{q}} = rac{3e^{r}}{5}
Cross-multiplying gives:
or:
Thus:
e^{q + r} = rac{25}{9}
Now, since and can take any real values, let's express in terms of :
If we solve for , we have:
e^{q} = rac{25}{9 e^{r}}
This leads us to:
3e^{q} = rac{75}{9 e^{r}}
However, observe that for both expressions and to be positive, must be positive. Consequently, there cannot be a scenario wherein , , and can hold as consecutive terms of a geometric sequence due to the inherent contradictions from equating the ratios.
Thus, we conclude that there is no possible value of for which , , and can be consecutive terms of a geometric sequence.