Question

The 25th term, 10th term and the last term of an ap are - 67, - 22 and - 82 respectively find the common difference and the number of terms​

Answer 1

Answer:

Step-by-step explanation:

Answer 2

The common difference of the given AP is $d=-3$.

The total number of terms in the given AP is $n=30$.

Given,

An AP with $n$ number of terms such that

$a_{25} = -67\\a_{10} =-22\\a_{n} =-82$

To find,

Common difference, $d$

Number of terms, $n$

Solution,

We know that the general term of an AP is given by,

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

Where a is the first term.

Using this,

$a_{25} = a+(25-1)d$      

⇒$a_{25} =a+24d$

⇒$a+24d=-67$                 ------ (i)

Similarly,

$a_{10} =a+9d$

⇒$a+9d=-22$                 -------(ii)

And,

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

⇒$a+(n-1)d=-82$       --------(iii)

Now subtracting equation (i) and (ii)

⇒$(a+24d)-(a+9d)=-67-(-22)$

⇒$a+24d-a-9d=-67+22$

⇒$15d=-45$

⇒$d=-3$

Now put this value in equation (ii) to get the value of $a$

⇒$a+9(-3)=-22$

⇒$a=-22+27$

⇒$a=5$

For the value of $n$, let us put the obtained value of $d$ and $a$ in equation (iii)

⇒$5+(n-1)(-3)=-82$

⇒$5+82=3(n-1)$

⇒$\frac{87}{3} =n-1$

⇒$n=29+1$

⇒$n=30$

Therefore,

Common difference of given AP is $d=-3$.

Number of terms of given AP is $n=30$.