Question

Find the sum of all even numbers between 1 and 150
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Answer 1

sum of all even numbers between 1 to 150 = 5700 .

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Answer 2

$\huge \mathbb{ANSWER}$

$\sf{The \: AP \: is \: 2 , 4, 6 ,..........150}$

$\sf{the \: first \: even \: number \: between \: 1 to 150=2}$

$\rm{a1 =2}$

$\sf{Last \: even \: number \: between \: 1 to 150=150}$

$\rm{an =150}$

$\sf{common difference (d)=second \: term-first \: term}$

$\rm{=4-2 = 2}</p><p>$

$\sf{we \: know \: that}$

$\sf{an=a+(n-1) d}$

$\sf{150= 2 + (n-1) ×2}$

$\sf{150-2 = (n-1) ×2}$

$\sf{148 = (n-1) ×2}$

$\sf{74 = n-1}$

$\sf{74 +1 = n}$

$\sf{75 = n}$

we know that

$\rm{sn = \frac{n}{2}(a + an)}$

$\sf{ \frac{75}{2}(2 + 150)}$

$\sf{ \frac{75}{2} = 150}$

$\sf{ \frac{75}{2} \times 152 \: 76}$

$\sf{75 \times 76}$

$\sf{5700}$

So, sum of all even number between 1 to 150 is 5700