Question

M^3-m^5+m^7-m^9...+m^19 найти Sn

Answer 1

Ответ:

это арифметическая прогрессия

(m^3+m^19)19/2

Answer 2

To find the sum of the series $S_n = M^3 - m^5 + m^7 - m^9 + \ldots + m^{19}$, we need to determine the pattern and properties of the series.

Looking at the series, we can observe that it consists of alternating terms with odd exponents of 'm'. The powers of 'm' in the series are: 3, 5, 7, 9, ..., 19. The difference between consecutive powers is 2.

The general form of the terms in the series can be written as follows:

$a_k = (-1)^{k+1} \cdot m^{2k+1}$

where 'k' is the index of the term, starting from 1.

Now, let's find the value of 'S_n' by adding all the terms up to the 'n'-th term:

$S_n = a_1 + a_2 + a_3 + \ldots + a_n$

Substituting the expression for $a_k$:

$S_n = (-1)^{2} \cdot m^{2+1} + (-1)^{3} \cdot m^{2\cdot2+1} + (-1)^{4} \cdot m^{2\cdot3+1} + \ldots + (-1)^{n+1} \cdot m^{2\cdot n +1}$

Now, we can factor out the common terms:

$S_n = (-1)^{2}\cdot m^{1} \cdot (1 + m^2 + m^4 + \ldots + m^{2n})$

Next, let's evaluate the sum of the geometric series $1 + m^2 + m^4 + \ldots + m^{2n}$. The sum of the geometric series is given by:

$1 + m^2 + m^4 + \ldots + m^{2n} = \frac{{m^{2n+2} - 1}}{{m^2 - 1}}$

Substitute this back into the expression for $S_n$:

$S_n = (-1)^2 \cdot m^1 \cdot \frac{{m^{2n+2} - 1}}{{m^2 - 1}}$

Simplifying further:

$S_n = m \cdot \frac{{m^{2n+2} - 1}}{{m^2 - 1}}$

So, the sum of the series is $S_n = m \cdot \frac{{m^{2n+2} - 1}}{{m^2 - 1}}$.