Question

The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms is 400. Find the number of terms and the common difference.​

Answer 1

Step-by-step explanation:

Given:-

• The first term of an AP is 5 and the last term is 45.

• The sum is 400.

To Find:-

• The number of terms

• The common difference.

Solution:-

Let the first term be " a "

And last term be " l " = aₙ

And number of terms be " n "

So, a = 5 and l = aₙ = 45

The formula for finding Sum of n terms is:-

$\boxed{\bf\dashrightarrow S_{n} = \dfrac{n}{2} \big(a + l\big)}$

Given, Sum of n terms = 400

$\rm \dashrightarrow\dfrac{n}{2} \big(a + l\big) = 400$

$\rm \dashrightarrow\dfrac{n}{2} \big(5 + 45\big) = 400$

$\rm \dashrightarrow\dfrac{n}{2} \times 50= 400$

$\rm \dashrightarrow 25n= 400$

$\rm \dashrightarrow n= \dfrac{400}{25}$

$\rm \dashrightarrow n= 16$

$\underline{\boxed{ \therefore\textsf{ \textbf{ \pink{Number \: of \: terms = 16}}}}}$

$\rm Since, \: l = a_{n} = a + (n - 1)d$

$\rm \dashrightarrow a + (n - 1)d = 45$

$\rm \dashrightarrow 5 + (16 - 1)d = 45$

$\rm \dashrightarrow 5 + 15d = 45$

$\rm \dashrightarrow 15d = 45 - 5$

$\rm \dashrightarrow 15d = 40$

$\rm \dashrightarrow d = \dfrac{40}{15}$

$\rm \dashrightarrow d = \dfrac{8}{3}$

$\underline{\boxed{ \bf{ \purple{\therefore Common \: difference = \frac{8}{3}}}}}$

Answer 2

Given : The first term of an AP is 5, the last term is 45 and the sum is 400.

To Find : Find the number of terms is 400 & Find the number of terms and the common difference ?

______________

Solution : Let the first term be " a " And last term be " l " = aₙ And number of terms be " n ".

$~$

We have,

• First term (a) = 5
• Last term (l) = 45
• Sum of the terms of the AP (Sn) = 400

$~$

We have, to find,

• Common diffrence (d) = ?
• Number of terms (n) = ?

$~$

Now, According to question :

$~$

• We will find the number of terms. So, for calculating number of terms we will use given below formula :

$~$

$\underline{\frak{\pmb{As ~we ~know~ that~:}}}$

• $\boxed{\frak{S_{n}~=~\dfrac{n}{2} \bigg(a ~+~ l\bigg)}}$★

$~$

$\underline{\frak{\pmb{According ~to ~the ~given~ question~:}}}$

$~$

$\qquad{\sf:\implies{400~=~\dfrac{n}{2} \bigg(5 ~+ ~45\bigg)}}$

$\qquad{\sf:\implies{400~=~\dfrac{n}{2} \bigg(50\bigg)}}$

$\qquad{\sf:\implies{n ~=~ \dfrac{(400)(2)}{50}}}$

$\qquad{\sf:\implies{n~ =~ \cancel\dfrac{800}{50}}}$

$\qquad:\implies{\underline{\boxed{\frak{\pink{\pmb{n~=~16}}}}}}$★

$~$

Therefore,

• Number of terms = 16.

___________________

• Now, we will find the common difference.To finding the common diffrence we have to use the below formula

$~$

$\underline{\frak{\pmb{As ~we ~know~ that~:}}}$

• $\boxed{\frak{l~=~a ~+~ \bigg(n ~-~ 1\bigg)d}}$★

$~$

$\underline{\frak{\pmb{According ~to ~the ~given~ question~:}}}$

$~$

$\qquad{\sf:\implies{45~ = ~5 ~+ ~\bigg(16 ~- ~1\bigg) ~d}}$

$\qquad{\sf:\implies{45 ~- ~5 ~=~ 15d}}$

$\qquad{\sf:\implies{40~ = ~15d}}$

$\qquad{\sf:\implies{d ~=~\dfrac{40}{15}}}$

$\qquad:\implies{\underline{\boxed{\frak{\pink{\pmb{d~=~\dfrac{8}{3}}}}}}}$★

$~$

Hence,

• $\sf{Common~difference~=~\bf{\pmb{\dfrac{8}{3}}}}$