MHB Sum of First 20 Terms of Arithmetic Progression with Even Terms Removed

[mathlearn]

First term of the progression is 3 & the common difference is 4

Find the sum of the first 20 terms of the progression that is obtained by removing the terms in the even positions of the given progressions, such as the second term,fourh term, sixth term.

Formula preferences

For the sum of an arithmetic progression I prefer,

$S_n=\frac{n}{2}\left\{2a+(n-1)d\right\}$

For the term of an arithmetic progression,

$T_n=a+(n-1)d$

Many Thanks :)

 

[Greg]

Hint: What is the difference between the first and third terms? What is the difference between the third and fifth terms?

 

[mathlearn]

Hint: What is the difference between the first and third terms? What is the difference between the third and fifth terms?

The third term

$T_3=3+(3-1)4=8+3=11$

The difference between the first and the third term is $3 {\underbrace{\phantom{2d) + (3e}}_{\text{+8}}} 11$

The fifth term

$T_5=3+(5-1)4=16+3=19$

The difference between the first and the third term is $11 {\underbrace{\phantom{2d) + (3e}}_{\text{+8}}} 19$

So there is a common difference of 8 between the terms

$S_{20}=\frac{20}{2}\left\{2*3+(20-1)8\right\}$
$S_{20}=10\left\{6+19*8\right\}$
$S_{20}=10\left\{6+152\right\}$
$S_{20}=10\left\{158\right\}$
$S_{20}=1580$

Correct ?

Many Thanks ;)

 

[Greg]

How many terms are there in the sequence of odd-numbered terms?

 

[mathlearn]

From the first 20 term the number of odd numbers would be 10. (Thinking)

 

[Greg]

So you've got an arithmetic sequence with ten terms. What is the first term? What is the common difference?

 

[mathlearn]

So you've got an arithmetic sequence with ten terms. What is the first term? What is the common difference?

The first term is 3 & the common difference as calculated above is +8.

Find the sum of the first 20 terms of the progression that is obtained by removing the terms in the even positions of the given progressions, such as the second term,fourh term, sixth term.

And I guess you are supposed to find the sum of first 20 terms of the above arithmetic progression in which all the 20 terms are odd numbers (Thinking)

 

[Greg]

The third term

$T_3=3+(3-1)4=8+3=11$

The difference between the first and the third term is $3 {\underbrace{\phantom{2d) + (3e}}_{\text{+8}}} 11$

The fifth term

$T_5=3+(5-1)4=16+3=19$

The difference between the first and the third term is $11 {\underbrace{\phantom{2d) + (3e}}_{\text{+8}}} 19$

So there is a common difference of 8 between the terms

$S_{20}=\frac{20}{2}\left\{2*3+(20-1)8\right\}$
$S_{20}=10\left\{6+19*8\right\}$
$S_{20}=10\left\{6+152\right\}$
$S_{20}=10\left\{158\right\}$
$S_{20}=1580$

Correct ?

Many Thanks ;)

That's correct! Good work!

(I initially misread the problem ... )