Geometric sequences & series

A geometric series, S, has second term 60

The common ratio of S is 0.2

Find the exact value of the sum of the first five terms of S

A different geometric series, T, has second term 60 and positive common ratio r

The sum to infinity of T is T_∞

Show that

Find the maximum value of r – r²

Hence find the range of possible values of T_∞

Fully justify your answer.

The nth term of a sequence is u_n

The sequence is defined by

Find an expression, in terms of p, for u₂

It is given that u₃ 382

Show that p satisfies the equation

It is given that the sequence is a decreasing sequence.

Find the value of u₄ and the value of u₅

The limit of u_n as n tends to infinity is L

Write down an equation for L

Find the value of L

Given that

Hence find

The area, A, bounded by the curve with equation y 2^x, the x-axis, the y-axis and the line x 4 is approximated using eight rectangles of equal width as shown in the diagram below.

Show that the exact area of the largest rectangle is

The areas of these rectangles form a geometric sequence with common ratio

Find the exact value of the total area of the eight rectangles. Give your answer in the form k(1 2) where k is a rational number.

More accurate approximations for A can be found by increasing the number, n, of rectangles used.

Find the exact value of the limit of the approximations for A as n

Find the sum to infinity of the sequence.

Hence, or otherwise, evaluate

Find the smallest positive exact value of , in radians, which satisfies the equation

A geometric sequence has a sum to infinity of 3

A second sequence is formed by multiplying each term of the original sequence by 2

What is the sum to infinity of the new sequence?

Circle your answer.

The sum to infinity does not 6 3 6 exist

A building has a leaking roof and, while it is raining, water drips into a 12 litre bucket.

When the rain stops, the bucket is one third full.

Water continues to drip into the bucket from a puddle on the roof.

In the first minute after the rain stops, 30 millilitres of water drips into the bucket.

In each subsequent minute, the amount of water that drips into the bucket reduces by 2%.

During the n th minute after the rain stops, the volume of water that drips into the

Find W₂

Explain why

and state the value of A.

Find the increase in the water in the bucket 15 minutes after the rain stops.

Give your answer to the nearest millilitre.

Assuming it does not start to rain again, find the maximum amount of water in the bucket.

After several hours the water has stopped dripping.

Give two reasons why the amount of water in the bucket is not as much as the found in part (d).

The sum to infinity of a geometric series is 96

The first term of the series is less than 30

The second term of the series is 18

Find the first term and common ratio of the series.

Show that the nth term of the series, u_n, can be written as

Hence show that

log₃ u_n n(1 2 log₃ 2) 5 log₃ 2

y e^x x x 16 (a) (sin cos)

Simplify your answer.

Hence, show that

where a is a rational number.

The areas of the finite regions bounded by the curve and the x-axis are denoted by A, A,..., A,...

Find the exact value of the area A₁

Show that

Given that

show that the exact value of the total area enclosed between the curve and the x-axis is

Helen is creating a mosaic pattern by placing square tiles next to each other along a straight line.

The area of each tile is half the area of the previous tile, and the sides of the largest tile have length w centimetres.

Find, in terms of w, the length of the sides of the second largest tile.

Assume the tiles are in contact with adjacent tiles, but do not overlap.

Show that, no matter how many tiles are in the pattern, the total length of the series of tiles will be less than 3.5 w.

Helen decides the pattern will look better if she leaves a 3 millimetre gap between adjacent tiles.

Explain how you could refine the model used in part (b) to account for the 3 millimetre gap, and state how the total length of the series of tiles will be affected.