Question

The first and last terms of an AP are 4 and 81 respectively. The common difference is 7.

(a) How many terms are there in the AP
(b) Find the sum?
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Answer 1

Step-by-step explanation:

$\textbf{\underline{Given,,}}</p><p>$

First term of AP,a=4

Last term of AP,an=81

Common difference,d=7

(a) To find number of terms in AP

We know,,

an=a+(n-1)d

=>81=4+(n-1)7

=>77/7=n-1

=>11+1=n

=>n=12

Hence there are 12 terms in the AP.

(b)Sum,

Sn=n/2[a+an]

=>Sn=12/2[4+81]

=>Sn=6×85

=>Sn=510

Hence,, sum of the all terms is 510.

Hope it helps ❣️❣️❣️

Answer 2

AnswEr:

$\sf{Given}\begin{cases}\sf{First\:Term\:(a)\:=\:4}\\ \sf{Last \: Term\:=\: 81} \\ \sf{Common\: Difference\:(d)\:=\:7}\end{cases}$

$\:\:\:\:\:\bold{\underline{\sf{\red{By\: Using\: Formula}}}}$

$\dag\:\:\small\boxed{\sf{\purple{a_{n}\:=\:a +(n\:-\:1)d}}}$

Putting Values

$:\implies\sf\: 81 = 4 + (n - 1)7$

$:\implies\sf\: 81 = 4 + 7n - 7$

$:\implies\sf\: 81 = -3 + 7n$

$:\implies\sf\: 81 + 3 = 7n$

$:\implies\sf\: 84 = 7n$

$:\implies\sf\:n = \cancel\dfrac{84}{7}$

$:\implies\large{\underline{\boxed{\sf{\red{n\:=\:12}}}}}$

$\bold{\underline{\sf{There\:are\:12\:Terms\:in\;AP.}}}$

$\rule{200}2$

Now, Finding Sum,

Formula:

$\dag\:\;\small\boxed{\sf{\purple{S_{n}\:=\: \frac{n}{2}[2a \:+\:(n\:-\:1)d]}}}$

Putting Values

$:\implies\sf\: \cancel\dfrac{12}{2} [2(4) +(12 - 1)7]$

$:\implies\sf\:6[ 8 + 77]$

$:\implies\sf\:6 \times 85$

$:\implies\large{\underline{\boxed{\sf{\red{S_{n}\:=\: 510}}}}}$

$\bold{\underline{\sf{Hence,\:Sum\:of\: Given\:AP\:is \:510.}}}$