Given that $\displaystyle \sum_{r=1}^n (4r + A) = \frac{n}{2}(4n - 6)$

where $\displaystyle A$ is a constant

(a) show that $\displaystyle A = -5$

Hence, or otherwise,

(b) (i) evaluate $\displaystyle \sum_{21}^{60} (4r - 5)$

(ii) find the greatest value of $\displaystyle n$ such that

An arithmetic series $\displaystyle F$ has third term 8 and fifth term 20

Given that the sum to $\displaystyle n$ terms of $\displaystyle F$ is greater than 220

find the least value of $\displaystyle n$

A geometric series $\displaystyle G$ has first term $\displaystyle a$ and common ratio $\displaystyle r$

The third term of $\displaystyle G$ is 10 and the seventh term of $\displaystyle G$ is 40

Given that the second term of $\displaystyle G$ is negative,

(a) find

(i) the exact value of $\displaystyle r$

(ii) the value of $\displaystyle a$

(b) Explain why $\displaystyle G$ does not have a sum to infinity.

(i) A geometric series $\displaystyle G$ has first term $\displaystyle (3 + \sqrt{3})$ and common ratio $\displaystyle r$

The second term of $\displaystyle G$ is $\displaystyle \sqrt{3}$

(a) Find, showing all your working, the exact value of $\displaystyle r$

Give your answer in the form $\displaystyle \frac{p + \sqrt{3}}{q}$ where $\displaystyle p$ and $\displaystyle q$ are integers to be found.

(b) Explain why $\displaystyle G$ is convergent.

(ii) A different geometric series $\displaystyle H$ has first term 8 and common ratio 0.6

The sum to $\displaystyle n$ terms of $\displaystyle H$ is $\displaystyle S_n$ and the sum to infinity of $\displaystyle H$ is $\displaystyle T$

Find, using logarithms, the least value of $\displaystyle n$ such that $\displaystyle T - S_n < 0.12$

(i) Find the value of $\displaystyle \sum_{k=6}^{17} 3(2)^{k-1}$

(ii) The common ratio of a geometric series $\displaystyle G$ is positive.

The third term of $\displaystyle G$ is $\displaystyle \frac{7}{13}$ and the ninth term of $\displaystyle G$ is $\displaystyle \frac{448}{9477}$

$\displaystyle G$ is convergent with sum to infinity $\displaystyle S$

Find the exact value of $\displaystyle S$

(a) Show that $\displaystyle \sum_{r=1}^n (5r - 3) = \frac{n}{2}(5n - 1)$

(b) Hence, or otherwise, evaluate $\displaystyle \sum_{r=31}^{60} (5r - 3)$

Given that $\displaystyle \sum_{r=1}^n (5r - 3) = 3783$

(c) find the value of $\displaystyle n$

The sum of the first and second terms of a geometric series $\displaystyle G$ is 400

The sum of the second and third terms of $\displaystyle G$ is 100

(a) Show that the common ratio of $\displaystyle G$ is $\displaystyle \frac{1}{4}$

(b) Show that the first term of $\displaystyle G$ is 320

(c) Find the sum to infinity of $\displaystyle G$

The sum to $\displaystyle n$ terms of $\displaystyle G$ is $\displaystyle S_n$

(d) Find, using logarithms, the least value of $\displaystyle n$ such that

The sum of the fifth, sixth and seventh terms of an arithmetic series $\displaystyle A$ is nine times the sum of the first and second terms.

The third term of $\displaystyle A$ is 12

(a) Find the first term and common difference of $\displaystyle A$

The $\displaystyle n$th term of $\displaystyle A$ is $\displaystyle u_n$

(b) Find the value of $\displaystyle \sum_{r=15}^{60} u_r$

The sum to $\displaystyle n$ terms of $\displaystyle A$ is $\displaystyle S_n$

Given that $\displaystyle 2S_n - 5u_n = 10$

(c) find the value of $\displaystyle n$

The sum of the first 10 terms of an arithmetic series $\displaystyle A$ is $\displaystyle 36k + 1$ where $\displaystyle k$ is a constant.

The 6th term of $\displaystyle A$ is $\displaystyle 4k + 1$

(a) (i) Find an expression in terms of $\displaystyle k$ for the common difference of $\displaystyle A$

(ii) Show that the first term of $\displaystyle A$ is $\displaystyle -8$

Given that the 4th term of $\displaystyle A$ is 7

(b) show that $\displaystyle k = 4$

The sum of the first $\displaystyle n$ terms of $\displaystyle A$ is $\displaystyle S_n$ and the $\displaystyle n$th term of $\displaystyle A$ is $\displaystyle U_n$

(c) Find the value of $\displaystyle n$ such that $\displaystyle S_n = 5U_{n+10} + 105$

(a) Use the factor theorem to show that $\displaystyle (4x - 1)$ is a factor of

(b) Hence, or otherwise, find the exact roots of the equation

A geometric series $\displaystyle G$ has first term $\displaystyle a$ and common ratio $\displaystyle r$
The third term of $\displaystyle G$ is 9 and the sum to infinity of $\displaystyle G$ is 192

(c) Show that $\displaystyle 64r^3 - 64r^2 + 3 = 0$

Given that $\displaystyle r$ is a rational number

(d) write down the value of $\displaystyle r$

(e) show that $\displaystyle a = 144$

The sum to $\displaystyle n$ terms of $\displaystyle G$ is $\displaystyle S_n$

(f) Using logarithms, find the least value of $\displaystyle n$ such that $\displaystyle S_n > 191.9$

The sum of the first 2 terms of a geometric series $\displaystyle G$ is 360
The sum of the 2nd and 3rd terms of $\displaystyle G$ is 288

The $\displaystyle n$th term of $\displaystyle G$ is $\displaystyle U_n$

(a) Show that $\displaystyle U_n = A \left( \frac{4}{5} \right)^{n-1}$ where $\displaystyle A$ is an integer to be found.

(b) Explain why $\displaystyle G$ is convergent.

(c) Hence find the sum to infinity of $\displaystyle G$

(d) Find the least number of terms for which the sum is greater than 978

A geometric series $\displaystyle G$ with common ratio $\displaystyle r$, has first term 16 and third term $\displaystyle \frac{2704}{625}$

(a) Find the two possible values of $\displaystyle r$

Given that $\displaystyle r > 0$

(b) find the sum to infinity of $\displaystyle G$

The sum to $\displaystyle n$ terms of $\displaystyle G$ is greater than 33

(c) Find, using logarithms, the least possible value of $\displaystyle n$

Show your working clearly.

The $\displaystyle n$th term of a geometric series $\displaystyle G$ is $\displaystyle U_n$ and the sum of the first $\displaystyle n$ terms of $\displaystyle G$ is $\displaystyle S_n$

Given that $\displaystyle U_n = \frac{25}{4} \left( \frac{3}{5} \right)^n$

(a) find the exact value of $\displaystyle U_5$

(b) Show that $\displaystyle S_n = \sum_{r=1}^n \frac{A}{B} \left( \frac{3}{5} \right)^{r-1}$ where $\displaystyle A$ and $\displaystyle B$ are integers to be found.

The sum to infinity of $\displaystyle G$ is $\displaystyle S$

(c) Find the least value of $\displaystyle n$ such that $\displaystyle S - S_n < 0.045$

The $\displaystyle n$th term of a convergent geometric series is $\displaystyle 8^{(1-2n)}$

Find the sum to infinity of the series.

Give your answer in the form $\displaystyle \frac{p}{q}$ where $\displaystyle p$ and $\displaystyle q$ are integers to be found.

The sum to $\displaystyle n$ terms of an arithmetic series $\displaystyle A$ is $\displaystyle S_n$

The sum of the first four terms of $\displaystyle A$ is 42 and the fifth term of $\displaystyle A$ is 23

(a) Show that $\displaystyle S_n = \sum_{r=1}^n (Pr - Q)$ where $\displaystyle P$ and $\displaystyle Q$ are prime numbers.

$\displaystyle S_{2n} - 3U_n = 1062$ where $\displaystyle U_n$ is the $\displaystyle n$th term of $\displaystyle A$

(b) Find the value of $\displaystyle n$

The common ratio of a geometric series $\displaystyle G$ is positive.

The sum of the first 4 terms of $\displaystyle G$ is $\displaystyle 80$.

The sum to infinity of $\displaystyle G$ is $\displaystyle 81$.

Show that the sum of the first 7 terms of $\displaystyle G$ differs from the sum to infinity of $\displaystyle G$ by

An arithmetic series has 5th term $\displaystyle 16$ and 100th term $\displaystyle 301$.

Find the sum of the first 50 terms of the series.

A geometric series has first term $\displaystyle a$ and common ratio $\displaystyle r$, where $\displaystyle r>0$.

Given that the 3rd term of the series is $\displaystyle 5$ and that the 5th term of the series is $\displaystyle \frac{5}{2}$,

(a) find

(i) the exact value of $\displaystyle r$,

(ii) the value of $\displaystyle a$.

(b) Find the sum to infinity of this series. Give your answer in the form

where $\displaystyle p$ and $\displaystyle q$ are integers.

The $\displaystyle n$th term of an arithmetic series is $\displaystyle u_n$, where

Given that the sum of the first $\displaystyle n$ terms of the series is $\displaystyle S_n$, show that

where $\displaystyle a$ is an integer whose value is to be found.

The $\displaystyle n$th term of a geometric series with common ratio $\displaystyle r$ is $\displaystyle u_n$.

Given that

(a) find the two possible values of $\displaystyle r$.

Given that the series is convergent with sum to infinity $\displaystyle S$,

(b) find the exact value of $\displaystyle S$.

An arithmetic series $\displaystyle A$ has first term $\displaystyle a$ and common difference $\displaystyle d$. The sum $\displaystyle S_n$ of the first $\displaystyle n$ terms of $\displaystyle A$ is given by

(a) Find the value of $\displaystyle a$ and the value of $\displaystyle d$.

(b) Find the 20th term of $\displaystyle A$.

Given that

(c) find the value of $\displaystyle p$.

The $\displaystyle n$th term of an arithmetic series is $\displaystyle u_n$ such that

where $\displaystyle a$ and $\displaystyle b$ are positive integers.

Given that $\displaystyle u_2=\mathrm{ln}\,12$ and that $\displaystyle u_5=\mathrm{ln}\,768$, find the value of $\displaystyle a$ and the value of $\displaystyle b$.

(a) Show that

(b) Hence evaluate

Given that

(c) find the value of $\displaystyle n$.

A geometric series has first term $\displaystyle (x-3)$, second term $\displaystyle (x+1)$ and third term $\displaystyle (4x-2)$.

(a) Find the two possible values of $\displaystyle x$.

Given that $\displaystyle x<1$,

(b) show that the series is convergent.

The sum to infinity of the series is $\displaystyle S$.

(c) Find the value of $\displaystyle S$.

The sum of the first $\displaystyle n$ terms of the series is $\displaystyle S_n$. Given that

(d) find the value of $\displaystyle n$.

The $\displaystyle n$th term of an arithmetic series $\displaystyle A$ is $\displaystyle a_n$. The $\displaystyle n$th term of a geometric series $\displaystyle G$ is $\displaystyle t_n$.

For these two series,

Find

(i) the common ratio of $\displaystyle G$,

(ii) the common difference of $\displaystyle A$.

The sum of the first $\displaystyle n$ terms of an arithmetic series is $\displaystyle S_n$.

Given that

(a) show that

The $\displaystyle r$th term of this arithmetic series is $\displaystyle t_r$.

Given that

(b) find the value of $\displaystyle n$.

(a) Show that

(b) Hence, or otherwise, find the least value of $\displaystyle n$ such that

Given that $\displaystyle S_n=n(2n-1)$, $\displaystyle t_n=4n-3$ and that

(c) find the value of $\displaystyle n$.

The sum of the first $\displaystyle n$ terms of an arithmetic series is An where

A $\displaystyle {\mathit{l}}_{n}\,=\,\sum_{r=1}^{n}\,(4r^{\star}+5)$

(a) For this arithmetic series, find

(i) the first term,

(ii) the common difference.

The sum of the first $\displaystyle n$ terms of a geometric series is Gn where

Gn $\displaystyle =\sum_{r=1}^{n}4(3)^{r-1}$

(b) For this geometric series, find

(i) the first term,

(ii) the common ratio.

(c) Find the value of $\displaystyle n$ for which A14 $\displaystyle - 6 =$ Gn

The A geometric nth term series of the has series first is term Un a and common ratio $\displaystyle r (r > 0)$

Given that $\displaystyle U1 + 3U2 = 8$ and that $\displaystyle U2 \times U3 = 4U5$

(a) find

(i) the value of $\displaystyle r$

(ii) the value of a

(b) Hence show that Un $\displaystyle ={\frac{2^{n+2}}{3^{n}}}$

(c) Find the least value of $\displaystyle n$ such that Un $\displaystyle < 0.05$

The nth term of a geometric series $\displaystyle G$ is un

The first term of $\displaystyle G$ is a and the common ratio of $\displaystyle G$ is $\displaystyle {\mathit{J}}_{\cdot}$ where $\displaystyle r\geqslant0$

Given that $\displaystyle u3 = 4$ and that $\displaystyle u7 = 16$

(a) (i) show that $\displaystyle U = 2$

(ii) find the value of a.

(b) Find the least value of $\displaystyle n$ for which un $\displaystyle 500$

The sum of the first $\displaystyle n$ terms of $\displaystyle G$ is Sn

(c) Find Give your answer in the form $\displaystyle p (1 + 2)$ where $\displaystyle p$ is an integer.