Question

an arithmetic progression has 10 terms the sum of the odd terms is 245 whereas the sum of the even terms is 305 find the common difference

Answer 1

Heya

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ist ten terms in A.P are

a , a + d , a + 2d , a + 3d , a + 4d , a + 5d , a + 6d , a + 7d , a + 8d And a + 9d

Where a is ist term and d is common difference.

ACCORDING TO THE QUESTION

a + a + 2d + a + 4d + a + 6d + a + 8d = 245

5a + 20d = 245

a + 4d = 49.... Equation ( i )

And

a + d + a + 3d + a + 5d + a + 7d + a + 9d = 305

5a + 25d = 305

a + 5d = 61.... Equation ii

SOLVE i and ii we get

d = 12

So, common difference is 12.

Answer 2

Step-by-step explanation:

Given : An arithmetic progression has $10$ terms the sum of the odd terms is $245$ whereas the sum of the even terms is $305$.

To find : The common difference(d).

Formula used : $nth$ term of an AP is,

$an=a+(n-1)*d$

a is the first term, d is the common difference of AP.

• Calculation for common difference(d)

We have 10 terms such as,

$a1,a2,a3,a4,a5,a6,a7,a8,a9,a10.$

By using formula

⇒  $an=a+(n-1)*d$

if we have to find the value of first term the value of n is 1 so the term is,

⇒  $a1=a+(1-1)d$    

⇒  $a1=a$

Similarly we can write for other terms.

Sum of odd terms $=245$

⇒  $a1+a3+a5+a7+a9=245$

by using formula we can write,

⇒ $[a+(1-1)d]+[a+(3-1)d]+[a+(5-1)d]+[a+(7-1)d]+[a+(9-1)d]=245$

⇒  $a+a+2d+a+4d+a+6d+a+8d=245$

⇒  $5a+20d=245$

taking 5 as common from equation,

⇒  $5(a+4d)=245$

dividing both sides by 5,

⇒  $a+4d=\frac{245}{5}$

⇒  $a+4d=49$                       ---(1)

Sum of even terms $=305$

⇒  $a2+a4+a6+a8+a10=305$

⇒ $[a+(2-1)d]+[a+(4-1)d]+[a+(6-1)d]+[a+(8-1)d]+[a+(10-1)d]=305$

⇒  $a+d+a+3d+a+5d+a+7d+a+9d=305$

⇒  $5a+25d=305$

⇒  $5(a+5d)=305$

⇒  $a+5d=\frac{305}{5}$

⇒  $a+5d=61$                   --(2)

Now we have two linear equations (1) and (2) and we can solve them by substitution method.

Extract a from equation (1),

⇒  $a=49-4d$                 ---(3)

substitute this a in equation (2),

⇒  $a+5d=61$      

⇒  $49-4d+5d=61$

⇒  $d=61-49$

⇒  $d=12$

now put this d in equation (3),

⇒  $a=49-4d$

       $=49-4*12$

       $=49-48$

       $=1$

In this way we get the values of first term(a) and common difference (d) as,

$a=1$         and    $d=12$

Hence the common difference of AP is 12.