Question

1×2+2×3+3×4+......+999×1000

Answer 1

Ответ:

999020

Пошаговое объяснение:

1×2+2×3+3×4+999×1000

2+6+12+999000

Answer 2

Ответ:

333333000

Пошаговое объяснение:

$1*2+2*3+3*4+...+n*(n+1) = \frac{n*(n+1)*(n+2)}{3}$

n = 999;

Поставив 999 вместо n получаем:

$\frac{999*(999+1)*(999+2)}{3} = 333333000$

Answer 3

The expression you've written seems to represent a sum of products where each term consists of the product of two consecutive numbers, ranging from 1 to 1000. In mathematical notation, it can be represented as:

1×2 + 2×3 + 3×4 + ... + 999×1000

This type of sum can be simplified using the formula for the sum of the first n natural numbers, which is given by:

1 + 2 + 3 + ... + n = n × (n + 1) / 2

In your case, the expression involves products of consecutive numbers, so we can rewrite it as follows:

1×2 + 2×3 + 3×4 + ... + 999×1000 = (1×2) + (2×3) + (3×4) + ... + (999×1000) = 2 + 6 + 12 + ... + 999000

Notice that each term in the series is a multiple of the form k(k+1), where k ranges from 1 to 999. We can factor out k from each term:

2 + 6 + 12 + ... + 999000 = 2(1) + 3(2) + 4(3) + ... + 999(998)

Now we have a sum of the form k(k+1) for k ranging from 1 to 999. Using the formula for the sum of the first n terms of this sequence:

1^2 + 2^2 + 3^2 + ... + n^2 = n(n + 1)(2n + 1) / 6

We can write the sum as:

2(1) + 3(2) + 4(3) + ... + 999(998) = 2 × 1^2 + 3 × 2^2 + 4 × 3^2 + ... + 999 × 998^2 = 2 × (1 × 2 × 3) + 3 × (2 × 3 × 4) + ... + 999 × (998 × 999 × 1000) / 6

= (2/6) × (1 × 2 × 3) + (3/6) × (2 × 3 × 4) + ... + (999/6) × (998 × 999 × 1000) = (1/3) × 6! + (1/2) × 7! + ... + (1/6) × (999 × 998 × 999 × 1000)

Calculating this sum will require a calculator or computer software due to its complexity. The result will be a large number.