Question

show that the sum of 5th term from beginning and 5th term from the end of an ap is an equal to the sum of first and last term

Answer 1

GENERAL EQUATION:X+(n-1) d

5th term from beginning:x+4d------(1)

" " from end: X+(n-5) d-----(2)

(1) +(2) =2X+dn-d (Frst half completed)

sum of first term and last term=

first term(x) +last term(x+(n-1) d)

2x+dn-d(second half also completed)

LHS=RHS

thus proved..

Answer 2

It is showed that the sum of 5th term from beginning and 5th term from the end of an Arithmetic Progression (A.P. )  is an equal to the sum of first and last term.

Step-by-step explanation:

Given:

The sum of 5th term from beginning and 5th term from the end of an A.P.  is an equal to the sum of first and last term

To Find:

It is to be showed that the sum of 5th term from beginning and 5th term from the end of an A.P. is an equal to the sum of first and last term.

Formula Used:

qth term of from begining of  the Arithmetic Progression  tp= m+(p-1)z ---- formula no.01.

qth term of from end of  the Arithmetic Progression  tpl= n-(p-1)z  ----formula  no.02

Where

m = first term of  Arithmetic Progression

n= last term of  Arithmetic Progression

z = common difference.

q = number of the terms

tq = qth term of the Arithmetic Progression .

Solution:

As given-the sum of 5th term from beginning and 5th term from the end of an A.P. is an equal to the sum of first and last term.

Applying formula no.01.

5th terms from beginning of A.P. $t_{5} =m+(5-1) z$

                                                       $t_{5} =m+4 z$  -------- equation no.01

Applying formula no.02.

5th terms from end of A.P $t_{5l} = n-(5-1)z$

                                                       $t_{5l} = n-4z$   -------- equation no. 01.

It is to be showed that the sum of 5th term from beginning and 5th term from the end of an A.P. is an equal to the sum of first and last term.

LHS =the sum of 5th term from beginning and 5th term from the end of an A.P.

       $=t_{5} +t_{5l}$

Putting the value of $t_{5} \ and\ &t_{5l}$ from equation 01 ad equation no.02.

        $=m+4 z+n-4z$

         $= m+n$

         =The sum of first and last term.

$LHS=RHS$    

Thus, It is  showed that the sum of 5th term from beginning and 5th term from the end of an Arithmetic Progression (A.P. )is an equal to the sum of first and last term.