Question

name 3 a consicutive number whose multiplication and addition is same​

Answer 1

$\sf\red{\underline{\underline{Answer:}}}$

$\sf{The \ three \ consecutive \ numbers \ are \ 1, \ 2 \ and \ 3}$

$\sf{or \ -3, \ -2 \ and \ -1.}$

$\sf\orange{Given:}$

$\sf{Three \ consecutive \ numbers \ have \ same }$

$\sf{multiplication \ and \ addition.}$

$\sf\pink{To \ find:}$

$\sf{The \ numbers.}$

$\sf\green{\underline{\underline{Solution:}}}$

$\sf{Let \ the \ constant \ be \ n.}$

$\sf{\therefore{The \ numbers \ are \ (n-1), \ n \ and \ (n+1).}}$

$\sf{According \ to \ the \ given \ condition. }$

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

$\sf{\therefore{3n=(n^{2}-1)\times \ n}}$

$\sf{\therefore{n^{2}-1=\frac{3n}{n}}}$

$\sf{\therefore{n^{2}-1=3}}$

$\sf{\therefore{n^{2}=3+1}}$

$\sf{\therefore{n^{2}=4}}$

$\sf{\therefore{n=\pm \ 2}}$

$\sf{If \ n=2, \ the \ numbers \ are }$

$\sf{n-1=2-1=1,}$

$\sf{n=2,}$

$\sf{n+1=2+1=3.}$

$\sf{If \ n=-2, \ the \ number \ are}$

$\sf{n-1=-2-1=-3,}$

$\sf{n=-2,}$

$\sf{n=-2+1=-1.}$

$\sf\purple{\tt{\therefore{The \ three \ consecutive \ numbers \ are \ 1, \ 2 \ and \ 3}}}$

$\sf\purple{\tt{or \ -1, \ -2 \ and \ -3.}}$

Answer 2

Answer:

Call the first number x. The second must be x+1, and the third x+2.

Set their sum and product equal: x + x+1 + x+2 = x (x+1) (x+2).

Work out both sides: 3x + 3 = x^3 + 3x^2 + 2x. It’s higher degree than a linear equation, so move all the terms to the same side: x^3 +3x^2 -x -3 = 0. So far, all basic algebra, but many students don’t learn how to solve this until they get to Pre-Calculus.

Consider the FIRST PAIR of terms separate from the SECOND PAIR, and factor those pairs SEPARATELY. Each pair has its own common factor: x^2 for the first pair, and -1 for the second pair. So, x^2 (X+3) -1(X+3) = 0. See that the factors of the original pairs EACH CONTAIN THE FACTOR (X+3). This itself is a COMMON FACTOR, so we can write (X+3) [x^2 - 1] = 0. Now we set each of the NEW factors equal to zero and solve: X+3 = 0 and x^2 - 1 = 0. Easy! Get -3, +1, -1.

Important: These are NOT the three numbers we seek. We solved for x, so each of these is a possibility for the FIRST number. So the answers are -3, -2, -1 AND +1, +2, +3 AND -1, 0, +1. Check … you’ll see that each triple’s sum is the same as its product!