Arithmetic Progression Questions (1 Viewer)

[acevipa]

I'm having trouble with a few arithmetic progression questions:

1. Find the sum of all integers between 20 and 50 that are divisible by 3.
2. Show that the sum of the first n odd natural numbers is a perfect square.
3. For the series functon defined by S<SUB>n </SUB>= 3n<SUP>2</SUP> – 11n, find t<SUB>n </SUB>and hence show that the sequence is arithmetic.

Would someone mind showing me how to do them, or maybe give me a hint as to how to solve them. That would be most appreciated. Thanks.

 

[Timothy.Siu]

I'm having trouble with a few arithmetic progression questions:

1. Find the sum of all integers between 20 and 50 that are divisible by 3.
2. Show that the sum of the first n odd natural numbers is a perfect square.
3. For the series functon defined by S<SUB>n </SUB>= 3n<SUP>2</SUP> – 11n, find t<SUB>n </SUB>and hence show that the sequence is arithmetic.

Would someone mind showing me how to do them, or maybe give me a hint as to how to solve them. That would be most appreciated. Thanks.

i dunno if i've learnt this but for the first one, divisible by 3 from there is 21,24,27,...,48
Tn=18+3n 48=18+3n n=10
therefore S10=(21+48)/2 x 10=345

dunno how to do 2nd one properly
3rd one..i'll try
For the series functon defined by Sn = 3n^2 – 11n, find tn and hence show that the sequence is arithmetic.
S1=T1=-8
S2=-10 S2-T1=-2=T2
S3=-6 S3-S2=4=T3 series is -8+-2+4.... common difference is 6
i dunno if thats enough to show its arithmetic

 

[gurmies]

For question 2, Odd terms are:

1, 3, 5, ..., 2n - 1

Sum to n terms:

n/2 (1 + 2n-1)

= n/2 (2n)

= n^2, which is a perfect square, as n is an integer (a positive one, not that it matters)

 

[Trebla]

I'm having trouble with a few arithmetic progression questions:

1. Find the sum of all integers between 20 and 50 that are divisible by 3.
2. Show that the sum of the first n odd natural numbers is a perfect square.
3. For the series functon defined by S<SUB>n </SUB>= 3n<SUP>2</SUP> – 11n, find t<SUB>n </SUB>and hence show that the sequence is arithmetic.

Would someone mind showing me how to do them, or maybe give me a hint as to how to solve them. That would be most appreciated. Thanks.

For the third one you have to show it is an arithmetic sequence for ALL n, not just the first few integers of n.
S_n = 3n² - 11n
T_n = S_n - S_{n - 1}
= 3n² - 11n - [3(n - 1)² - 11(n - 1)]
= 3n² - 11n - 3n² + 6n - 3 + 11n - 11
= 6n - 14
For an arithmetic series, the relation: T_n - T_{n - 1} = T_{n + 1} - T_n must hold and equal a constant (common difference)
T_n - T_{n - 1} = 6n - 14 - [6(n - 1) - 14]
= 6n - 14 - 6n + 6 + 14
= 6
T_{n + 1} - T_n = 6(n + 1) - 14 - (6n - 14)
= 6n + 6 - 14 - 6n + 14
= 6
Hence we have an arithmetic series/sequence...