Question

find the sum 3 +11+19+......+803

Answer 1

$\huge{\bf{\underline{\purple{Solution :}}}}$

Given,

• In this question First term (a) = 3
• Common difference (d ) = 11 - 3 = 8

To Find ,

• The sum of 3 + 11 + 19 + ........ + 803 = ?

Solution,

We know that :

$\boxed{\bold{\red{n^{th} \ term = a + (n - 1) d}}}$

Here ,

• a is the first term .
• n is the number of terms .
• d is the difference between two Consecutive terms .

⇒If 803 is nth term of given A.P then,

Therefore,

$\implies \bold{a_{n} = a + (n - 1) d}$

∵ $\bold{a_{n} = 803}$

∴ $\implies \bold{a + (n-1) d = 803}$

$\implies \bold{ 3 + (n-1) 8 = 803}$

$\implies \bold{(n - 1) 8 = 803 -3}$

$\implies \bold{ n - 1 = \frac{800}{8}}$

$\implies \bold{ n - 1 = 100}$

$\implies \boxed{\bold{ n = 101}}$

So, AP has 101 terms

∴ $\bf{\bol{\pink{Sum \ of \ n \ Terms}}}$

We know that :

$\implies \boxed{\bold{s_{n} = \frac{n}{2} [ 2a + ( n - 1)d]}}$

$\implies \bold{s_{100} = \frac{101}{2} [ 2 \times + ( 101 - 1) \times 8]}$

$\implies \bold{s_{101} = \frac{101}{2} [ 6 + 800 ]}$

$\implies \bold{s_{101} = \frac{101 \times 806}{2}}$

$\implies \bold{101 \times 403}$

$\implies \boxed{\bold{\red{40703}}}$ ( Answer )

$\rule{200}2$

Answer 2

$\tt{\red{Given \: - }}$

• a = 3

• d = 11 - 3 = 8

We know that :

$</p><p>\tt{\boxed{\bold{\purple{n^{th} \ term = a + (n - 1) d}}}}$

$\tt{\orange{\implies {\boxed{bold{a_{n} = a + (n - 1) d}}}}}$

$\implies \boxed{\red{s_{n} = \frac{n}{2} [ 2a + ( n - 1)d]}}$

Hence

${\tt{\green{ \therefore{a + (n-1) d = 803}}}}$

Placing Values And Substituting :

$\implies \boxed{\bold{ n = 101}}$

So, AP has 101 terms

$\implies \boxed{\bold{\red{S_{n}40703}}}........(A)$