Question

Find the sum of APs of first 20 terms of the series 1+2+3------?

Answer 1

Given :-

A.P. = 1 , 2 ,3 - - - - - - -

Required to find :-

• Sum of the first 20 terms in the Arithmetic progession ?

Formulae used :-

$\large{\dagger{\boxed{\rm{ {a}_{nth} = a + ( n - 1 ) d }}}}$

$\large{\dagger{\boxed{\rm{ {S}_{nth} = \dfrac{n}{2} [first \; term + last \; term ] }}}}$

Solution :-

Given A.P

1,2,3 - - - - - -

We need to find the sum of the first 20 terms .

So,

The First term = 1

Common difference =( Second term - First term )=( Third term - Second term )

=> ( 2 - 1 ) = ( 3 - 2 )

=> ( 1 ) = ( 1 )

Hence,

Common difference = 1

We need to find the 20th term of the A.P. in order to find the sum of the first 20 terms .

So, using the formula

$\large{\dagger{\boxed{\rm{ {a}_{nth} = a + ( n - 1 ) d }}}}$

Here,

a = First term

d = common difference

n = The term number which you want to find

Hence,

$\Rightarrow{\rm{ {a}_{nth} = {a}_{20}}}$

So,

$\longrightarrow{\tt{ {a}_{20} = 1 + ( 20 - 1 ) 1 }}$

$\longrightarrow{\tt{ {a}_{20} = 1 + (19) 1 }}$

$\longrightarrow{\tt{ {a}_{20} = 1 + 19 }}$

$\longrightarrow{\tt{ {a}_{20} = 20 }}$

Hence

20th term = 20

Now let's find the sum of the first 20th term

using the formula,

$\large{\dagger{\boxed{\rm{ {S}_{nth} = \dfrac{n}{2} [first \; term + last \; term ] }}}}$

Here,

a = first term

d = common difference

n = The number till which you want to find the sum

So,

$\rm{\Rightarrow{ {S}_{nth} = {S}_{20}}}$

Here,

$\longrightarrow{\tt{ {S}_{20} = \dfrac{20}{2} [ 1 + 20 ] }}$

$\longrightarrow{\tt{ {S}_{20} = \dfrac{20}{2} [ 21 ] }}$

$\longrightarrow{\tt{ {S}_{20} = 10 \times 21 }}$

$\longrightarrow{\red{\tt{ {S}_{20} = 210 }}}$

Therefore,

$\large{\text{ Sum of first 20 terms = 210 }}{\bigstar}$