Question

x-y/x+y+3x-2y/x+y+5x-3y/x+y upto 11 terms ( arithmatic progression)​

Answer 1

Given to Find

The sum of an A.P up to 11 terms, where the given A.P is $\dfrac{x-y}{x+y} ,\ \dfrac{3x-2y}{x+y} ,\ \dfrac{5x-3y}{x+y} ,\ ...$.

Solution

① Finding the expression of the given A.P.

To find the first and last term, we need the expression of the given A.P.

Concepts,

where,

• $\bold{a_{n}}$ is the n-th term
• $\bold{a}$ is the first term
• $\bold{d}$ is the common difference
• $\bold{n}$ is the term number

The expression for the n-th term:

$\implies \bold{a_{n}=a+(n-1)d}$

Returning to the solution,

$\implies a_{n}=\dfrac{x-y}{x+y} +(\dfrac{2x-y}{x+y} )(n-1)$

$\implies a_{n}=\dfrac{x-y+2nx-ny-2x+y}{x+y}$

$\implies \boxed{\bold{a_{n}=\dfrac{(2n-1)x-ny}{x+y}}}$

② Finding the first and the last term.

Using the previously found $\bold{a_{n}}$, we can find the first and the last term.

Concepts,

where,

• $\bold{S_{n}}$ is the sum of first n terms
• $\bold{n}$ is the term number
• $\bold{a}$ is the first term
• $\bold{l}$ is the last term

Returning to the solution.

The first term is,

$\implies a=\dfrac{x-y}{x+y}$

The last term is,

$\implies l=\dfrac{21x-11y}{x+y}$

③ Conclusion.

Concepts,

where,

• $\bold{S_{n}}$ is the sum of first n terms
• $\bold{n}$ is the term number
• $\bold{a}$ is the first term
• $\bold{l}$ is the last term

The sum up to n-th term:-

$\implies \bold{S_{n}=\dfrac{n(a+l)}{2}}$

The sum up to 11 term is,

$\implies S_{11}=\dfrac{11}{2} (\dfrac{x-y}{x+y} +\dfrac{21x-11y}{x+y} )$

$\implies S_{11}=\dfrac{11}{2} (\dfrac{22x-12y}{x+y} )$

$\therefore \boxed{\bold{S_{11}=\dfrac{121x-66y}{x+y}}}$

This is the required answer.

More Information

Derivation of the Gauss sum

We are going to find the Gauss sum.

① The property of an A.P.

$\implies S_{n}=a+(a+d)+(a+2d)+...+(l-2d)+(l-d)+l$

$\implies S_{n}=l+(l-d)+(l-2d)+...+(a-2d)+(a-d)+a$

② Adding the two equations.

$\implies 2S_{n}=(a+l)+(a+l)+(a+l)+...+(a+l)+(a+l)+(a+l)$

$\implies \bold{\therfore S_{n}=\dfrac{n(a+l)}{2}}$