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# If x is a positive integer greater than 1, what is the sum of the mult

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Re: If x is a positive integer greater than 1, what is the sum of the mult [#permalink]
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Ans :- B.
I substituted 3 in all answer choices.
Option B & C were both satisfying the condition.

Only B option satisfied.

Can anyone tell a simpler way to solve.
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Re: If x is a positive integer greater than 1, what is the sum of the mult [#permalink]
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Bunuel wrote:
If x is a positive integer greater than 1, what is the sum of the multiples of x from x to x^2, inclusive?

(A) x(x + 1)(x-1)
(B) x^2(x + 1)/2
(C) x^2(x-1)
(D) (x^3 + 2x)/2
(E) x(x-1)^2

Kudos for a correct solution.

We have to find the sum of multiples of x from x to x^2 inclusive

let x = 3
then sum of the multiples from 3 to 9 will be
3+6+9
or 3(1+2+3)

let x=5
then sum of multiples from 5 to 25 will be
5+10+15+20+25
or 5(1+2+3+4+5)

From these values it is clear that th sum of multiples of x from x to x^2 inclusive will be
x(sum of first x natural numbers)
or $$x*$$ $$\frac{x(x+1)}{2}$$
or $$\frac{x^2(x+1)}{2}$$

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Re: If x is a positive integer greater than 1, what is the sum of the mult [#permalink]
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Solution:

Sum = x + 2x +3x +.. + x(x) = x(1+2+3...+x) = x((x/2)(2 +(x-1)) = (x^2)(x+1)/2
So, option B
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Re: If x is a positive integer greater than 1, what is the sum of the mult [#permalink]
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Bunuel wrote:
If x is a positive integer greater than 1, what is the sum of the multiples of x from x to x^2, inclusive?

(A) x(x + 1)(x-1)
(B) x^2(x + 1)/2
(C) x^2(x-1)
(D) (x^3 + 2x)/2
(E) x(x-1)^2

Kudos for a correct solution.

Let, x=2

sum of the multiples of x from x (i.e. 2) to x^2 (i.e. 2^2=4), inclusive = 2+4 = 6

@x=2
(A) x(x + 1)(x-1) = 2*3*1 = 6
(B) x^2(x + 1)/2 = 2^2*3/2 = 6

(C) x^2(x-1) = 2^2*3 = 12
(D) (x^3 + 2x)/2 = (8+4)/2 = 6
(E) x(x-1)^2 = 2*1 = 2

i.e. Possible Options are only A, B and D

Now, Let, x=3

Sum of the multiples of x from x (i.e. 3) to x^2 (i.e. 3^2=9), inclusive = 3+6+9 = 18

@x=3
(A) x(x + 1)(x-1) = 3*4*2 = 24
(B) x^2(x + 1)/2 = 3^2*4/2 = 18
(D) (x^3 + 2x)/2 = (27+6)/2 = 33/2

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Re: If x is a positive integer greater than 1, what is the sum of the mult [#permalink]
Here is what i did in this one =>Putting x=5
Sum = 5+10+15+20+25=75
only option that matches is B.

Hence B.
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Re: If x is a positive integer greater than 1, what is the sum of the mult [#permalink]
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Bunuel wrote:
If x is a positive integer greater than 1, what is the sum of the multiples of x from x to x^2, inclusive?

(A) x(x + 1)(x-1)
(B) x^2(x + 1)/2
(C) x^2(x-1)
(D) (x^3 + 2x)/2
(E) x(x-1)^2

Since we know that x is a positive integer greater than 1, we can let x = 2. Thus, we need to determine the sum of multiples of 2 from 2 to 4 inclusive. We see that the sum of the multiples of 2 is 2 + 4 = 6. Now we need to determine which of the answer choices is equivalent to 6:

A) x(x + 1)(x - 1) = 2(3)(1) = 6….YES

B) x^2(x + 1)/2 = 4(3)/2 = 6….YES

C) x^2(x - 1) = 4(1) = 4….NO

D) (x^3 + 2x)/2 = (8 + 4)/2 = 6….YES

E) x(x - 1)^2 = 2(1)^2 = 2….NO

To decide between the answer choices A, B, and D, we can let x = 3. The multiples of 3 between 3 and 9 inclusive are 3, 6, and 9. We have 3 + 6 + 9 = 18. Let’s plug x = 3 into answer choices A, B, and D and see which one(s) produce 18:

A) x(x + 1)(x - 1) = 3(4)(2) = 24….NO

B) x^2(x + 1)/2 =9(4)/2 = 18….YES

D) (x^3 + 2x)/2 = (27 + 6)/2 = 16.5….NO

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Re: If x is a positive integer greater than 1, what is the sum of the mult [#permalink]

Either through trial and error, or through

$$a_n=a_1+(n-1)d \implies x^2=x+(n-1)x$$

you'll find that $$n=x$$

therefore, since we are dealing with an equally spaced set,

$$average*n=sum \implies \frac{(x+x^2)}{2}*x= \frac{(1+x)x^2}{2}$$
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Re: If x is a positive integer greater than 1, what is the sum of the mult [#permalink]
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Re: If x is a positive integer greater than 1, what is the sum of the mult [#permalink]
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