# mathematics

1 ) Use the arithmetic sequence of numbers 1 , 3 , 5 , 7 , 9 . to find the following

a ) What is d , the difference between any 2 terms

Answer : d 2

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d 3-1 5-3 7-5 9-7 2

b ) Using the formula of the nth term of an arithmetic sequence , what is the 101st term

Answer : nth term 201

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an a (n-1 ) d , where

a 1st term 1

d difference between 2 terms 2

n 24

an 1 (101-1 [banner_entry_middle]

)2 1 100 (2 1 200 201

c ) Using the formula for the sum of an arithmetic series , what is the sum of the first 20 terms

Answer : Sn 400

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Sn n /2 (a1 an , where

n 20

a1 1st term 1

an nth term

an 1 (20-1 )2 1 (19 )2 1 83 39

Sn 20 /2 (1 39 10 (40 400

d ) Using the formula for the sum of an arithmetic series , what is the sum of the first 30 terms

Answer : Sn 900

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Sn n /2 (a1 an , where

n 20

a1 1st term 1

an nth term

an 1 (30-1 )2 1 (29 )2 1 58 59

Sn 30 /2 (1 59 15 (60 900

e ) What observation can you make about these sums of this series (HINT : It would be beneficial to find a few more sums like the sum of the first 2 , then the first 3 , etc ? Express your observations as a general formula in n

Answer : Sn (n )[2a (n-1 )d] /2

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The sum of terms increases as more terms are added . The sum of “n ” terms is n times the average of the 1st and the nth term

Sn (n (a an /2 , where

a 1st term

an a (n-1 ) d

Sn (n (a an /2 (n )[a a (n-1 )d] /2 (n )[2a (n-1 )d] /2

2 ) Use the geometric sequence of numbers 1 , 2 , 4 , 8 . to find the following

a ) What is r , the ratio between 2 consecutive terms

Answer : r 2

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r 2 /1 4 /2 8 /4 2

b ) Using the formula for the nth term of a geometric sequence , what is the 24th term

Answer : an 8388608

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an a1rn-1 , where

n 24

a1 1st term 1

r ratio between 2 terms 2

an 1 224-1 223 8388608

c ) Using the formula for the nth term of a geometric sequence , what is the sum first 10 terms

Answer : Sn 1023

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Sn a1 (rn-1 / r-1 , where

n 10

a1 1st term 1

r ratio between 2 terms 2

Sn 1 (210 -1 (2-1 1023

3 ) Use the geometric sequence of numbers 1 , 1 /2 , 1 /4 , 1 /8 . to find the following

a ) What is r , the ratio between 2 consecutive terms

Answer : r 1 /2

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**Author:** Essay Raptor