Increasing sequences
Before the Aviva Stadium sells out, Aoife notices that the crowd at each Leinster home game seems to grow a little every week. "Seems" is not good enough for a mathematician: in this extension you will learn to prove that a sequence is increasing for every $n$, once and for all, without working out the terms one by one. At the end we will come back to Aoife's crowds.
The difference criterion
For a sequence given by a formula $T_n$ (the $n$th term), $T_{n+1}$ means "replace every $n$ by $n+1$". An arithmetic sequence $T_n = a + (n-1)d$ has $T_{n+1}-T_n = d$, so it is strictly increasing exactly when the common difference $d > 0$.
For a series, the partial sum is $S_n = T_1 + T_2 + \dots + T_n$, so $S_{n+1} - S_n = T_{n+1}$. If every $T_{n+1} > 0$, then $S_n$ is strictly increasing.
Write down the first four terms of the sequence $a_n = 3n - 5$ and check, by calculating the difference $a_{n+1} - a_n$, that it is a strictly increasing sequence.
Write down the first four terms of the sequence $a_n = 3n - 5$ and check, by calculating the difference $a_{n+1} - a_n$, that it is a strictly increasing sequence.
A student decides to save money for a trip. In the first month he saves $10$ euros, and each subsequent month he saves $5$ euros more than the previous month. Write down…
A student decides to save money for a trip. In the first month he saves $10$ euros, and each subsequent month he saves $5$ euros more than the previous month. Write down the general term for the money saved in month $n$ and justify mathematically why this sequence is strictly increasing.
Prove algebraically that the sequence with general term $a_n = \frac{3n-1}{2n+5}$ is strictly increasing. To do so, expand the inequality $a_{n+1} - a_n > 0$ and justify…
Prove algebraically that the sequence with general term $a_n = \frac{3n-1}{2n+5}$ is strictly increasing. To do so, expand the inequality $a_{n+1} - a_n > 0$ and justify the signs of the resulting numerator and denominator.
Given the sequence defined as the partial sum $S_n = \sum_{i=1}^n \frac{1}{i^2} = 1 + \frac{1}{4} + \frac{1}{9} + \dots + \frac{1}{n^2}$, justify analytically why this…
Given the sequence defined as the partial sum $S_n = \sum_{i=1}^n \frac{1}{i^2} = 1 + \frac{1}{4} + \frac{1}{9} + \dots + \frac{1}{n^2}$, justify analytically why this sequence is strictly increasing and deduce whether the difference between consecutive terms increases or decreases as $n$ becomes very large.
The ratio criterion
When all the terms are positive, it is more convenient to compare them by dividing rather than subtracting.
For a geometric sequence $T_n = ar^{n-1}$ with $a>0$, the ratio is the common ratio $r$, so it is strictly increasing when $r>1$. For a sequence such as $\left(1+\frac1n\right)^n$, expand with the binomial theorem: $\left(1+\frac1n\right)^n = \sum_{r=0}^{n}\binom{n}{r}\frac{1}{n^r}$, where the term with $r=2$ is $\frac{n(n-1)}{2n^2}=\frac12\left(1-\frac1n\right)$. Each such term grows with $n$, and $a_{n+1}$ has one extra positive term.
A sequence is bounded above if there is a number $M$ with $a_n \le M$ for all $n$. An increasing sequence may still be bounded above: it can keep rising without ever passing $M$.
Given the geometric sequence $d_n = 3^n$, use the quotient method (that is, calculate $\frac{d_{n+1}}{d_n}$) to prove that it is a strictly increasing sequence.
Given the geometric sequence $d_n = 3^n$, use the quotient method (that is, calculate $\frac{d_{n+1}}{d_n}$) to prove that it is a strictly increasing sequence.
The population of a bacteria colony in a controlled culture is modelled by the sequence $P_n = 200 \times (1,15)^n$, where $n \geq 1$ represents the number of days…
The population of a bacteria colony in a controlled culture is modelled by the sequence $P_n = 200 \times (1,15)^n$, where $n \geq 1$ represents the number of days elapsed. Use the ratio test $\frac{P_{n+1}}{P_n}$ to prove rigorously that the population is constantly growing.
In an industrial heating process, the temperature of a metal alloy at minute $n$ follows the law $T_n = 120 - 90(0,75)^n$ in degrees Celsius. Prove mathematically that…
In an industrial heating process, the temperature of a metal alloy at minute $n$ follows the law $T_n = 120 - 90(0,75)^n$ in degrees Celsius. Prove mathematically that the temperature is strictly increasing. From the graph, deduce whether the sequence is bounded above.
Study the monotonicity of the sequence $x_n = \frac{2^n}{n!}$. Show that it is not strictly increasing for all $n$, indicating from which specific term onwards it begins…
Study the monotonicity of the sequence $x_n = \frac{2^n}{n!}$. Show that it is not strictly increasing for all $n$, indicating from which specific term onwards it begins to strictly decrease.
Consider the sequence $a_n = \left(1 + \frac{1}{n}\right)^n$. Given that this sequence models the growth of capital under 100% annual compound interest with $n$…
Consider the sequence $a_n = \left(1 + \frac{1}{n}\right)^n$. Given that this sequence models the growth of capital under 100% annual compound interest with $n$ compounding periods (whose limit is continuous compounding), calculate its first three terms to check empirically that it is increasing. Then, set up the start of its general proof by expanding the binomial theorem for $a_n$ and $a_{n+1}$.
Sequences with radicals
With roots, the difference $a_{n+1}-a_n$ usually gives an apparent indeterminate form, which is resolved by multiplying and dividing by the conjugate, in the same way as when you rationalise a denominator containing a surd.
For a sequence defined by recurrence such as $u_{n+1}=\sqrt{c+u_n}$ with $u_n>0$, compare squares: $u_{n+1}^2-u_n^2 = c+u_n-u_n^2$. If this is positive, then $u_{n+1}>u_n$, because both terms are positive.
Consider the sequence with radicals $c_n = \sqrt{n+5}$. Prove that it is strictly increasing. To avoid working directly with inequalities of surds, multiply and divide…
Consider the sequence with radicals $c_n = \sqrt{n+5}$. Prove that it is strictly increasing. To avoid working directly with inequalities of surds, multiply and divide the expression $c_{n+1} - c_n$ by its conjugate.
Prove that the sequence with general term $a_n = \sqrt{n^2+n} - n$ is strictly increasing. Hint: Multiply and divide by the conjugate expression before analysing the…
Prove that the sequence with general term $a_n = \sqrt{n^2+n} - n$ is strictly increasing. Hint: Multiply and divide by the conjugate expression before analysing the difference $a_{n+1} - a_n$.
A server network records the mean latency time (in milliseconds) using the recurrence sequence $u_1 = 2$, $u_{n+1} = \sqrt{6 + u_n}$. Given that all generated terms are…
A server network records the mean latency time (in milliseconds) using the recurrence sequence $u_1 = 2$, $u_{n+1} = \sqrt{6 + u_n}$. Given that all generated terms are positive and strictly less than $3$, prove that the sequence is strictly increasing by analysing the sign of $u_{n+1}^2 - u_n^2$.
Sequences with a parameter
Some problems ask you to find for which values of a parameter a sequence is increasing: you set up the condition $a_{n+1}-a_n>0$ (or the ratio $>1$) and solve for the parameter.
Find the values of the parameter $k \in \mathbb{R}$ for which the sequence defined by $a_n = (k-3)n + 5$ is strictly increasing. Reason your answer from the difference…
Find the values of the parameter $k \in \mathbb{R}$ for which the sequence defined by $a_n = (k-3)n + 5$ is strictly increasing. Reason your answer from the difference $a_{n+1} - a_n$.
Find the range of values of $x \in \mathbb{R}$ for which the geometric sequence $a_n = (2x - 1)^n$ is strictly increasing and has exclusively positive terms.
Find the range of values of $x \in \mathbb{R}$ for which the geometric sequence $a_n = (2x - 1)^n$ is strictly increasing and has exclusively positive terms.
Analyse the growth of the sequence $a_n = \frac{n^2+k}{n}$, where $k$ is a real number. Prove that the sequence is strictly increasing for all $n \geq 1$ if and only if…
Analyse the growth of the sequence $a_n = \frac{n^2+k}{n}$, where $k$ is a real number. Prove that the sequence is strictly increasing for all $n \geq 1$ if and only if the parameter $k$ satisfies the condition $k < 2$.
Summary
Vuelve al principio: Aoife's crowd numbers are strictly increasing if $T_{n+1}-T_n>0$ for every $n$, and now you can prove it.