Multiples of 3.

Algebra Level 1

Find x x .

3 + 6 + 9 + . . . + 99 = x \large{3+6+9+...+99=x}


The answer is 1683.

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2 solutions

Blan Morrison
Jan 24, 2018

Since all of the numbers have a common factor of 3, we can re-write this sum as 3 ( 1 + 2 + 3 + . . . + 31 + 32 + 33 ) = x 3(1+2+3+...+31+32+33)=x Now, we can just find the 33rd triangle number, which can be found using the formula: 3 ( 33 ( 33 + 1 ) 2 ) 3(\frac{33(33+1)}{2}) This can be simplified to the expression of 99 × 17 99\times17 which is equivalent to 1 , 683 \boxed{1,683} .

Relevant wiki: Arithmetic Progressions

It is an arithmetic progression with a common difference of 3 3 . So the number of terms is 99 3 = 33 \dfrac{99}{3}=33 . And the desired sum is

x = 33 2 [ 2 ( 3 ) + ( 33 1 ) ( 3 ) ] = 1683 x=\dfrac{33}{2}[2(3)+(33-1)(3)]=\boxed{1683}

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