Question

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

Answer 1

GIVEN:

• First term (a) = 5
• Last term (an) = 45
• Sum of AP = 400

TO FIND:

• What is the number of terms and common difference ?

SOLUTION:

Let the number of terms be 'n'

We know that the formula for finding the sum of an AP is:-

$\large{\boxed{\bf{\star \: S_n = \dfrac{n}{2}[a + a_n] \: \star}}}$

According to question:-

$\sf{\longmapsto 400 = \dfrac{n}{2} [5 + 45]}$

$\sf{\longmapsto 400 \times 2 = n(50) }$

$\sf{\longmapsto 800 = 50n }$

$\sf{\longmapsto \cancel\dfrac{800}{50} = n }$

$\bf{\longmapsto 16 = n }$

• number of terms = n = 16

Let the common difference be 'd'

To find the Common difference, we use the formula:-

$\large{\boxed{\bf{\star \: a_n = a +(n-1)d \: \star}}}$

On putting the given values in the formula, we get

$\sf{\longmapsto 45 = 5+(16-1)d }$

$\sf{\longmapsto 45-5 = 15d}$

$\sf{\longmapsto 40 = 15d }$

$\sf{\longmapsto \cancel\dfrac{40}{15} = d}$

$\bf{\longmapsto \dfrac{8}{3} = d}$

• Common difference = d = 8/3

❝ Hence, the common difference and number of terms of an AP is 8/3 and 16 respectively. ❞

______________________

Answer 2

Solution :

$\underline{\bf{Given\::}}}}$

• First term of an A.P. (a) = 5
• Last term (l) = 45
• Sum, (Sn) = 400

$\underline{\bf{Explanation\::}}}}$

$\bigstar$Using formula of the last term of an A.P. to get number of term's :

$\boxed{\bf{S_n=\dfrac{n}{2} \bigg\lgroup a+l\bigg\rgroup}}}$

$\longrightarrow\tt{400=\dfrac{n}{2} \bigg\lgroup 5 + 45\bigg\rgroup}\\\\\longrightarrow\tt{400\times 2=n(50)}\\\\\longrightarrow\tt{800=50n}\\\\\longrightarrow\tt{n=\cancel{800/50}}\\\\\longrightarrow\bf{n=16}$

∴ Number of terms (n) = 16

$\bigstar$Using formula of the sum of an A.P. to get common difference (d) :

$\boxed{\bf{S_n=\frac{n}{2}\bigg[2a+(n-1)d\bigg]}}}$

• a is the first term
• d is the common difference.
• n is the term of an A.P.
• Sn is the sum of the A.P.

$\longrightarrow\tt{400=\cancel{\dfrac{16}{2}} \bigg[2(5)+(16-1)d\bigg]}\\\\\longrightarrow\tt{400=8[10+15d]}\\\\\longrightarrow\tt{\cancel{400/8}=10+15d}\\\\\longrightarrow\tt{50=10+15d}\\\\\longrightarrow\tt{15d=50-10}\\\\\longrightarrow\tt{15d=40}\\\\\longrightarrow\tt{d=\cancel{40/15}}\\\\\longrightarrow\bf{d=8/3}$

Thus;

The common difference (d) will be 8/3 .