Practice Questions

Define a sequence in mathematics.

Calculate the Geometric Mean (G.M.) between the numbers 4 and 16.

What is the primary difference between a sequence and a series?

Identify the first term and the common ratio of the Geometric Progression: $3, -6, 12, \ldots$.

Explain the difference between a finite sequence and an infinite sequence, providing one example for each.

Describe the method to determine if a given sequence is a Geometric Progression.

A sequence is defined by the formula $a_n = \frac{(-1)^n n^2}{n+1}$. Calculate the value of the 4th term, $a_4$.

Propose a method to determine if three given non-zero numbers $x, y, z$ are in G.P. without explicitly calculating the common ratio.

Calculate the sum of the first 8 terms of the geometric progression $3, 6, 12, \ldots$.

The first term of a Geometric Progression is $\sqrt{2}$ and the common ratio is also $\sqrt{2}$. Calculate the 15th term of this progression.

State the formula for the $n^{\text{th}}$ term of a Geometric Progression (G.P.).

Prove that if $a, b, c$ are in G.P., then their logarithms, $\log a, \log b, \log c$, are in A.P. (Assume $a,b,c > 0$ and the logarithm base is positive and not equal to 1).

Summarize the relationship between the Arithmetic Mean (A.M.) and Geometric Mean (G.M.) for any two positive real numbers.

Explain the procedure to derive the formula for the sum of the first $n$ terms of a Geometric Progression, $S_n$. Describe the two distinct cases based on the value of the common ratio $r$.

Write the series $3 + 9 + 27 + \ldots + 3^n$ using sigma notation.

How many terms of the geometric series $2 + 6 + 18 + \ldots$ are required to obtain a sum of 728?

Insert four numbers between 5 and 160 so that the resulting sequence is a Geometric Progression.

The Arithmetic Mean (A.M.) between two positive numbers is 25 and their Geometric Mean (G.M.) is 15. Calculate the two numbers.

The sum of $n$ terms of a sequence is given by $S_n = 5n^2 - 3n$. Justify that the sequence is an A.P. and formulate a rule for its $k^{th}$ term.

Define the Geometric Mean (G.M.) of two positive numbers $a$ and $b$.

Find the first five terms of the sequence whose $n^{\text{th}}$ term is given by $a_n = \frac{n^2 - 1}{n+2}$.

Given the sequence $5, 10, 20, 40, \ldots$, identify the first term '$a$', the common ratio '$r$', and write the formula for its $n^{\text{th}}$ term, $a_n$.

Describe the Fibonacci sequence. List its first 8 terms and explain the recurrence relation that generates it.

If the numbers $x-1, x+1,$ and $x+4$ are the first three terms of a G.P., calculate the value of $x$.

The first term of a G.P. is 81 and its fourth term is 3. Calculate the common ratio.

In a Geometric Progression, the 4th term is 54 and the 7th term is 1458. Calculate the first term and the common ratio.

Justify whether a sequence of non-zero numbers can be both an Arithmetic Progression (A.P.) and a Geometric Progression (G.P.) simultaneously.

Three numbers whose sum is 15 are in A.P. If 1, 4, and 19 are added to them respectively, the resulting numbers are in G.P. Create a system of equations to find the original numbers and justify your solution.

A ball is dropped from a height of $H$ meters. Each time it strikes the ground, it bounces back to $\frac{3}{4}$ of the height from which it fell. Formulate a model for the total vertical distance traveled by the ball until it comes to rest and evaluate this distance for $H=16$ meters.

List the first four terms of the sequence defined by the recurrence relation $a_1 = 4$ and $a_n = 2a_{n-1} - 1$ for $n > 1$.

The sum of the first three terms of a G.P. is 21 and their product is 216. Analyze this information to find the three terms.

Critique the statement: "The geometric mean of two distinct positive numbers is always smaller than their arithmetic mean."

If $a, b, c$ are in A.P. and $x, y, z$ are in G.P., derive an expression for $x^{b-c} y^{c-a} z^{a-b}$ and evaluate it.

Design a G.P. for which the sum of the first three terms is 26 and their product is 216. Find the first term and the common ratio.

Formulate an expression for the product $P$ of the first $n$ terms of a G.P. with first term $a$ and common ratio $r$. Justify your formula.

If the sum of the first $p$ terms of an A.P. is $q$ and the sum of the first $q$ terms is $p$, prove that the sum of the first $(p+q)$ terms is $-(p+q)$. Assume $p \neq q$.

Formulate a recurrence relation for a sequence where each term, starting from the third, is the sum of all preceding terms. Let the first term be $a_1=1$.

Justify why the sum of an infinite G.P. with a common ratio $|r| \geq 1$ (and first term $a \neq 0$) does not converge to a finite value.

Let $S_1, S_2, S_3, \dots, S_p$ be the sums of $p$ infinite geometric series whose first terms are $1, 2, 3, \dots, p$ respectively, and whose common ratios are $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots, \frac{1}{p+1}$ respectively. Create a formula for $S_k$ and then prove that $S_1 + S_2 + S_3 + \dots + S_p = \frac{p(p+3)}{2}$.

The sum of three numbers in a G.P. is 28. If we subtract 1, 1, and 5 from these numbers in that order, the resulting numbers form an A.P. Find the numbers in the G.P.

A ball is dropped from a height of 81 meters. Each time it hits the ground, it rebounds to $\frac{2}{3}$ of the height from which it fell. Calculate the total vertical distance traveled by the ball up to the moment it hits the ground for the 5th time.

Derive the condition for the roots of the cubic equation $x^3 - px^2 + qx - r = 0$ to be in G.P. and justify your derivation.

Calculate the sum to $n$ terms of the series $0.4 + 0.44 + 0.444 + \ldots$.

Summarize the concept of a series. Explain what is meant by the 'sum of a series' and how it differs from the series itself. Provide an example of a finite series and state its sum.

The product of the first three terms of a G.P. is 1000. If we add 6 to its second term and 7 to its third term, the new terms form an A.P. Analyze the given conditions to find the G.P.