Question

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

Answer 1

EXPLANATION.

=> First term of an Ap = 5

=> Last term of an Ap = 45

=> The sum = 400

To find number of terms and common

difference.

$\sf \to \: as \: we \: know \: that \\ \\ \sf \to \: s_{n} = \dfrac{n}{2} (a + l) \\ \\ \sf \to \: put \: the \: value \: in \: equation \\ \\ \sf \to \: 400 = \frac{n}{2} (5 + 45) \\ \\ \sf \to \: 400 = 25n \\ \\ \sf \to \: n \: = 16$

$\sf \to \: { \underline{some \: related \: formula}} \\ \\ \sf \to \: 1) = \: n \: terms \: of \: an \: ap \\ \\ \sf \to \: a_{n} = a + (n - 1)d \\ \\ \sf \to \: 2) = sum \: of \: nth \: terms \: of \: an \: ap \\ \\ \sf \to \: s_{n} = \frac{n}{2}(2a + (n - 1)d)$

$\sf \to \: nth \: term \: of \: an \: gp \\ \\ \sf \to \: t_{n} = ar {}^{n - 1} \\ \\ \sf \to \: sum \: of \: nth \: terms \: of \: an \: gp \\ \\ \sf \to \: s_{n} = \dfrac{a( {r}^{n} - 1)}{(r - 1)}$

Common difference

=> Sn = n/2 ( 2a + ( n - 1 ) d

=> 400 = 16/2 ( 2(5) + 15d )

=> 400 = 8 ( 10 + 15d )

=> 400 = 80 + 120d

=> 320 = 120d

=> d = 320 / 120 = 8/3

Answer 2

Solution :

We have,

• First term (a) = 5

• Last term (l) = 45

• Sum of the terms of the AP (Sn) = 400

And we have to find :

• Common diffrence (d) = ?

• Number of terms (n) = ?

Now, According to question :

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

★ Sn = n/2 (a + l) ★

• Now, Putting the values in the given formula we get :]

➳ 400 = n/2 (5 + 45)

➳ 400 = n/2 (50)

➳ n = (400)(2)/50

➳ n = 800/50

➳ n = 16

___________________

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

★ l = a + (n - 1)d ★

• Putting the values in above formula we get :]

➣ 45 = 5 + (16 - 1) d

➣ 45 - 5 = 15d

➣ 40 = 15d

➣ d = 40/15

➣ d = 8/3

_________________________

$\boxed{\underline{\underline{\bigstar \: \bf\:Extra\:Brainly\:knowlegde\:\bigstar}}}$

Que : What is Arithmetic Progression ?

Ans : It is a mathematical sequence in which difference is between two consecutive terms is always constant is called as$\sf \underline{\red {Arithmetic \: Progression.}}$

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$\boxed{\underline{\underline{\bigstar\:\bf\:Some \: important \: Formulae \: related \: to \: it \:\bigstar}}} \\ \\ </p><p></p><p>$

$\sf (i) \: The\: nth\: term\: of \:an \:AP:\:\:\:\: \red{a_n = a + (n - 1) d}$

$\sf(ii) \: Sum \: of \: n \: terms \: in \: AP :\:\:\:\: \purple{S = \dfrac{n}{2}\big\{ 2a + (n-1) d\big\}}$

$\sf (iii ) \: Sum \: of \: all \: terms \: in \: a \: finite \: AP \: with \: the \: last \: term \: as \: 'l': \:\:\:\: \pink{\dfrac{n}{2} (a + l)}$