Series, Progressions and the Binomial Expansion: Mathematics 9709 (Cambridge International AS & A Level)

In the binomial expansion of $(1 + 2x)^6$, what is the coefficient of $x^3$?

A small open-air theatre has rows of seats arranged so that each row has more seats than the row in front of it, forming an arithmetic progression. Row $1$ (nearest the stage) has $14$ seats, and each subsequent row has $3$ more seats than the row before it.

(a) Find the number of seats in Row $20$. [2]

(b) Find the total number of seats in the first $20$ rows. [2]

(c) The theatre has $30$ rows in total. Find the total seating capacity of the theatre. [2]

The amplitude of successive oscillations of a plucked guitar string decreases geometrically. The second oscillation has amplitude $18$ mm and the third oscillation has amplitude $12$ mm.

(a) Find the common ratio of the progression and the amplitude of the first oscillation. [3]

(b) Find the sum of the amplitudes of the first $10$ oscillations, giving your answer correct to $1$ decimal place. [3]

(c) Explain why the sum of the amplitudes of all the oscillations converges to a finite value as the number of oscillations increases without bound, and find this sum to infinity. [2]

(a) Find the expansion of $(1-2x)^6$ in ascending powers of $x$, up to and including the term in $x^3$. [4]

(b) Hence find the coefficient of $x^3$ in the expansion of $(2+5x)(1-2x)^6$. [3]

The numbers $4$, $k$ and $9$, in that order, are consecutive terms of a geometric progression, where $k>0$.

What is the value of $k$?

In the binomial expansion of $\left(x - \dfrac{2}{x^2}\right)^6$, what is the term that is independent of $x$ (the constant term)?

A runner is training for a marathon. In week $1$ of her training plan she runs $5$ km, and in each following week she runs $1.5$ km more than the week before, so that her weekly distances form an arithmetic progression.

(a) Find the distance she runs in week $15$. [2]

(b) Find her total training distance over the first $15$ weeks. [2]

(c) Find the least number of complete weeks needed for her total training distance to exceed $300$ km. [3]

A geometric progression has first term $8$ and sum to infinity $20$.

(a) Find the common ratio $r$. [2]

(b) Find the $4$th term of the progression. [2]

(c) Find the least value of $n$ for which the sum of the first $n$ terms exceeds $19.9$. [3]

In the binomial expansion of $(1+x)^n$, where $n$ is a positive integer, the coefficient of $x^2$ is $45$. What is the value of $n$?

A theme park runs a $6$-day flash sale on annual passes. On day $1$ of the sale, $800$ passes are sold. On each following day, the number of passes sold is half the number sold the day before, so that the daily sales figures form a geometric progression.

(a) Find the number of passes sold on day $5$. [2]

(b) Find the total number of passes sold over all $6$ days of the sale. [2]