Help with AP & GP questions~ (1 Viewer)

[Follz21]

Hi all

Just came across these questions in a few trial papers and can't seem to figure them out. Any help will be very much appreciated.

1) Show that √18 is the third term of an AP with begins as √2, 2√2,....

2) For any sequence, the sum of the first n terms is given by:

S(sub)n = n² - 5n

a) Find an expression for the nth term
b) Find the value of the tenth term.

3) A ball is dropped from a height of ten metres. It continually bounces back up to a height equal to four-fifths of its previous height. Find the total distance traveled by the ball.

4) For the series √2 + 1 + 1/√2 + ...

a) Find the sum to ten terms of the series
b) what is the difference between the sum to ten terms and the sum to infinity?

Thanks for any help!!

 

[x jiim]

Hi all

Just came across these questions in a few trial papers and can't seem to figure them out. Any help will be very much appreciated.

1) Show that √18 is the third term of an AP with begins as √2, 2√2,....

As it's an arithmetic progression, T3-T2 should be the same as T2-T1.
T2-T1 = 2√2 - √2 = √2
therefore T3 = T2 + √2 = 2√2 + √2 = 3√2 = 18

2) For any sequence, the sum of the first n terms is given by:

Sn = n² - 5n

a) Find an expression for the nth term
b) Find the value of the tenth term.

a) Sub in the values for 1, 2, 3. You get S1=-4, S2=-6, S3=-6
S1=T1=-4
S2=T1+T2=T2-4=-6 T2=-2
S3=T1+T2+T3 = T3-2-4=-6 T3=0
Which is obviously an AP, first term -4, difference 2
Tn= 2n-6

b) Sub n=10 into above formula
S10 = 2*10 -6 = 14.

3) A ball is dropped from a height of ten metres. It continually bounces back up to a height equal to four-fifths of its previous height. Find the total distance traveled by the ball.

Leave out the 10m for now- this is a GP with first term 2*8=16 and common ratio 4/5. You're looking for the sum to infinity of this, which is given by the equation a/(1-r), where a is the first term and r is the common ratio.
d= 10 + 16/(1/5)
= 10+80
=90m.
I think.

4) For the series √2 + 1 + 1/√2 + ...

a) Find the sum to ten terms of the series
b) what is the difference between the sum to ten terms and the sum to infinity?

Thanks for any help!!

a) It's another GP, first term √2, common ratio 1/√2. The formula for the sum of these things is
Sn= a(1- r^n)/(1-r)
so S10= √2(1-(1/√2)^10)/(1-(1/√2))
Which you then number bash to give whatever answer that is.

b) Sum to infinity given as before by a/(1-r)
=√2/(1-(1/√2))
= 1/(√2-1)
I think.

Hope that helps =\