Question

The first term of an AP is 5 and last term is 62. The sum of the terms is 670. Find the number of terms and common difference in the AP​

Answer 1

Given,  

First term (a) = 5

Last term (l) = 62

Sum of the terms (S) = 670

We know the equation:

$\boxed{\bold{\sf{S=\dfrac{n}{2}(a+l)}}}$  

Applying values to the equation:

$\sf{\implies 670=\dfrac{n}{2}(5+62)}$

$\sf{\implies 670=\dfrac{67n}{2}}$

$\sf{\implies 670\times2=67n}$

$\sf{\implies 1340=67n}$

$\sf{\implies n=\dfrac{1340}{67}}$

$\sf{\implies n=20}$

$\boxed{\bold{\sf{\therefore Number \ of \ terms=20}}}$

We know the equation:

$\boxed{\bold{\sf{\l=a+(n-1)d}}}$

Applying the values to equation:

$\sf{\implies 62=5+(20-1)d}$

$\sf{\implies 62=5+19d}$

$\sf{\implies 62-5=19d}$

$\sf{\implies 19d=57}$

$\sf{\implies d=\dfrac{57}{19}}$

$\sf{\implies d=3}$

$\boxed{\bold{\sf{\therefore Common \ Difference=3}}}$

 

Answer 2

Given:

The first term of an AP= 5, last term= 62, Sum of terms= 670.

To find:

The number of terms and the common difference in the AP.

Solution:

In an AP, we can find the sum (S) of all terms of the AP by using the formula:

$S = \frac{n(a + l)}{2}$

where n= number of terms, a= first term and l= last term.

Now, we have

a = 5, l = 62, S = 670

Applying the above formula, we have

$670 = \frac{n(5 + 62)}{2}$

$n = \frac{670 \times 2}{67}$

$n = 20$

So, the number of terms in the AP is 20.

Now, the difference of terms in the AP is obtained by:

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

Putting l = 62, a = 5 and n = 20 in above, we have

$62 = 5 + (20 - 1)d$

$57 = 19d$

$d = 3$

So, the difference of terms in the AP is 3.

Hence, the number of terms in the AP is 20 and the difference of terms in the AP is 3.