Thread: [SOLVED] Arithmetic and Geometric progressions

1. [SOLVED] Arithmetic and Geometric progressions

Having a bit of trouble when it comes to these questions, any help is very much appreciated

Q1. Programming languages such as miranda and Q use the notations [a..b] and [a,b..c] to represent finite sequences. For example, [1..5] is the sequence{1,2,3,4,5} and [1,3..11] is the sequence {1,3,5,7,9,11} (ie the 3 is the second term) For each of the notations [a..b] and [a,b..c], give a formula for the number of terms n of the sequence in terms of a,b and c.

Q2. (a) Are the terms of the sequence Un = {log2,log2^2,log2^3} in arithmetic or geometric progression? What about Vn={(log2),(log2)^2,(log2)^3}?
(b) Find the sum of the first 10 terms of both Un and Vn.

2. Originally Posted by Webby

Q2. (a) Are the terms of the sequence Un = {log2,log2^2,log2^3} in arithmetic or geometric progression? What about Vn={(log2),(log2)^2,(log2)^3}?
(b) Find the sum of the first 10 terms of both Un and Vn.

(a) $\displaystyle U_n= log2,2log2,3log2,...$ , d=log 2 , thus it is an AP .

$\displaystyle V_n=(log2),(log2)^2,(log2)^3,...$ , r=log2 , it is a GP .

They are both different .

(b) Use the formulas of sum of AP and GP respectively .

3. Hello, Webby!

Q2. (a) Are the terms of the sequence $\displaystyle U_n \:=\:\log2,\;\log(2^2),\:\log(2^3),\:\hdots$
in arithmetic or geometric progression?

We have: .$\displaystyle \log 2,\:2\log2,\:3\log2,\:\hdots$

This is an arithmetic sequence with first term, $\displaystyle a = \log 2$,
. . and common difference, $\displaystyle d = \log 2$

(b) What about: $\displaystyle V_n\:=\:(\log2),\:(\log2)^2,\:(\log2)^3,\:\hdots$

This is a geomtric sequence with first term, $\displaystyle a = \log 2$
. . and common ratio, $\displaystyle r = \log 2$

(c) Find the sum of the first 10 terms of both $\displaystyle U_n$ and $\displaystyle V_n.$

For $\displaystyle U_n\!:\;\;S_{10} \:=\:\frac{10}{2}\bigg[2\log 2 + 9\log2\bigg] \:=\:55\log2$

For $\displaystyle V_n\!:\;\;S_{10} \:=\:(\log2)\frac{1-(\log 2)^{10}}{1-\log2}$