Question

$\bf{Topic: \: Arithmetic\: Progression}$

The first term of the A.P. is 5, the last term is 45 and the sum of all its terms is 400. Find the number of terms and the common difference of the A.P.

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Answer 1

Answer:

Given :

first term of A.P = 5

the last term , l = 45

Sum of all its term ,S = 400

To Find :

☆ Number of terms

☆ common difference

Solution:

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

$\implies \: 400 = \frac{n}{2} (5 + 45) \\ \implies 400 \times 2 = n(50) \\ \implies \: 800 = 50n \\ \implies \: n \: = \frac{800}{50} = 16$

_____________________

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

$\implies \: 45 = 5 + (16 -1 )d \\ \implies \: 45 - 5 = 15d \\ \implies \: 40 = 15d \\ \implies \: d = \frac{40}{15} = \frac{8}{3} = 2.6$

So,the number of terms :

$\green{ \boxed{n = 16}}$

Common difference:

$\orange{ \boxed{d \: = \frac{8}{3} }}$

Answer 2

Answer:

${\large{\sf{\pmb{\underline{\red{Given:-}}}}}}$

• ● The first term of the A.P. is 5.
• ● The last term is the A.P. is 45.
• ● The sum of all its terms is 400.

$\large\sf{\pmb{\underline{\red{To \: Find:-}}}}$

• ● The number of terms
• ● The common difference of the A.P.

$\large\sf{\pmb{\underline{\red{Using \: Formula :-}}}}$

$\dag{\underline{\boxed{\sf{S_n= {\dfrac{n}{2}\big(a + l \big)}}}}}$

Where

• ● $\rm{S_n}$ = sum of all terms
• ● n = number of terms
• ● a = first term
• ● l = last term

$\rule{200}2$

${\dag{\underline{\boxed{\sf{l=a \big(n - 1 \big)d}}}}}$

Where

• ● l = Last term
• ● n = number of term
• ● d = common difference

$\large\sf{\pmb{\underline{\red{Solution :-}}}}$

${\underline{\pmb{\frak{Finding \: the \: number \: of \: term}}}}$

$: \implies{\sf{S_n= \bf{\dfrac{n}{2}\big(a + l \big)}}}$

• ● Substituting the values

$: \implies{\sf{400= \bf{\dfrac{n}{2}\big(5 + 45 \big)}}}$

$: \implies{\sf{400= \bf{\dfrac{n}{2}\big(50 \big)}}}$

$: \implies{\sf{400= \bf{\dfrac{n}{2} \times 50 }}}$

$: \implies{\sf{400= \bf{\dfrac{50n}{2}}}}$

$: \implies{\sf{400 \times 2= \bf{50n}}}$

$: \implies{\sf{800= \bf{50n}}}$

$: \implies{\sf{\dfrac{800}{50} = \bf{n}}}$

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

$: \implies{\sf{n} = \bf{16}}$

${\dag{\underline{\boxed{\sf{\pink{\pmb{n = 16}}}}}}}$

• ● Henceforth,The Number of term is 16.

$\rule{200}2$

${\underline{\pmb{\frak{Finding \: the \:common \: difference \: of \: A.P.}}}}$

${: \implies{\sf{l= \bf{a + \big(n - 1 \big)d}}}}$

• ● Substituting the values

${: \implies{\sf{45= \bf{5 + \big(16 - 1 \big)d}}}}$

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

${: \implies{\sf{40= \bf{15d}}}}$

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

${: \implies{\sf{\cancel{\dfrac{40}{15}} = \bf{d}}}}$

${: \implies{\sf{d} =\bf{\dfrac{8}{3} }}}$

${\dag{\underline{\boxed{\sf{\pink{\pmb{d =\dfrac{8}{3} }}}}}}}$

• ● Henceforth,The common difference of the A.P is 8/3.

$\large\sf{\pmb{\underline{\red{Know \: More :-}}}}$

★ Formula to find the numbers of term of an AP:

$: \implies \sf{n= \bigg[ \dfrac{(l - a)}{d} \bigg]}$

★ Formula to find the tsum of first n terms of an AP:

$: \implies\sf{S_n= \dfrac{n}{2} (a + l)}$

★ Formula to find the sum of squares of first n natural numbers of an AP:

$: \implies \sf{S = \dfrac{n(n + 1)(2n + 1)}{6} }$

★ Formula to find the nth term of an AP is the square of the number of terms:

$: \implies \sf{S = {n}^{2} }$

★ Formula to find the sum of of an AP:

$: \implies \sf{S = n(n+1)}$