Question

how are you
xxsinglequeenxX28
harder this time
solve this question and get 100 points
100 points for this simple question
1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25+26+27+28+29+30+31+32+33+34+35+36+37+38+39+40+41+42+43+44+45+46+47+48+49+50​​+51+52+53+54+55+56+57+58+59+60+61+62+63+64+65+66+67+68+69+70+71+72+73+74+75+76+77+78+79+80+81+82+83+84+85+86+87+88!89+90+91+92+93+94+95+96+97+98+99+100

harder this time​

Answer 1

Answer:

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Step-by-step explanation:

5050

The sum of all natural numbers from 1 to 100 is 5050.

Answer 2

$\large\underline{\sf{Solution-}}$

Given series is

$\rm :\longmapsto\:1 + 2 + 3 + - - - + 100$

Its an AP series with

$\rm :\longmapsto\:a = 1$

$\rm :\longmapsto\:d = 1$

$\rm :\longmapsto\: {n}^{th} \: term = 100$

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

aₙ is the nᵗʰ term.

a is the first term of the sequence.

n is the no. of terms.

d is the common difference.

Tʜᴜs,

$\rm :\longmapsto\:100 = 1 + (n - 1) \times 1$

$\rm :\longmapsto\:100 = 1 + n - 1$

$\bf\implies \:n = 100$

Now,

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of n  terms of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} \bigg(2 \:a\:+\:(n\:-\:1)\:d \bigg)}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

Sₙ is the sum of n terms of AP.

a is the first term of the sequence.

n is the no. of terms.

d is the common difference.

So, on substituting the values, we get

$\rm :\longmapsto\:S_n = \dfrac{100}{2} \bigg(2 \times 1 + (100 - 1) \times 1\bigg)$

$\rm :\longmapsto\:S_n =50 \times \bigg(2 + 100 - 1\bigg)$

$\rm :\longmapsto\:S_n =50 \times \bigg(101\bigg)$

$\bf\implies \:S_n = 5050$

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Short Cut

Sum of first n natural numbers is

$\boxed{\tt{ \displaystyle\sum_{n=1}^n\rm n \: = \: \frac{n(n \: + \: 1)}{2} \: }}$

So,

$\red{\rm :\longmapsto\:\displaystyle\sum_{n=1}^{100}\rm n = \frac{100(100 + 1)}{2} = 5050}$