Question

The first term of ap is 5 and last is 62 and the sun of there term is 670.FIND (i) The number of terms in the AP is
(a) 24 (b) 20 (c) 30 (ii) The common difference is
(a) 2 (b) 4 (c) 3(d) 5

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:

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

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

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

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

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

$\sf{\implies n=20}$

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

We know the equation:

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

Applying values to the equation:

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

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

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

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

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

$\sf{\implies d=3}$

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

Answer 2

Given: The first term of an AP is 5, the last is 62 and the sum of their terms is 670.

To find:

(i) The number of terms in the AP.

(ii) The common difference of the AP.

Solution:

An Arithmetic Progression (AP) is a sequence of numbers such that every consecutive number has a common difference.

(i)

• If the first and the last term of an arithmetic progression are given, the sum of the terms is calculated using the given formula.

        $S = \frac{n}{2} [a+l]$

• Here, S is the sum of all the terms, n is the number of terms in the arithmetic progression, a is the first term and l is the last term.
• On replacing the terms with the values given in the question,

        $670 = \frac{n}{2} [5+62]$

        $n = 20$

(ii)

• If the first and the last term of an arithmetic progression are given, the common difference is calculated using the given formula.

        $l = a + (n-1) d$

• Here, l is the last term of the arithmetic progression.

        $62 = 5 + (20 - 1) d$

        $d = 3$

Therefore,

(i) the number of terms in the arithmetic progression is 20.

(ii) the common difference of the arithmetic progression is 3.