3. Calcula las siguientes sumas. a. $$ \sum_{i=5}^{10} 2 i $$ b. $$ \sum_{j=0}^{10} 5^{j} $$ c. $$ \sum_{k=1}^{6}(3 k-2) $$ d. $$ \sum_{m=10}^{15} m^{2} $$ e. $$ \sum_{i=1}^{6}(-1)^{i+1} i $$ f. $$ \sum_{j=1}^{6}(-1)^{j} j $$

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**a. Cálculo de** $$ \sum_{i=5}^{10} 2i $$

Podemos factorizar la constante $$2$$:
$$
\sum_{i=5}^{10} 2i = 2 \sum_{i=5}^{10} i
$$
La suma de los enteros de $$5$$ a $$10$$ es:
$$
5+6+7+8+9+10 = 45
$$
Entonces:
$$
2 \times 45 = 90
$$

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**b. Cálculo de** $$ \sum_{j=0}^{10} 5^{j} $$

Esta es una suma de una progresión geométrica. La fórmula para la suma de una serie geométrica es:
$$
S = \frac{a\,(r^{n+1}-1)}{r-1}
$$
donde $$a=5^{0}=1$$, $$r=5$$ y el número de términos es $$n+1 = 11$$. Entonces:
$$
S = \frac{5^{11}-1}{5-1} = \frac{5^{11}-1}{4}
$$
Calculamos $$5^{11}$$:
$$
5^{11} = 48828125
$$
Por lo que:
$$
S = \frac{48828125-1}{4} = \frac{48828124}{4} = 12207031
$$

---

**c. Cálculo de** $$ \sum_{k=1}^{6}(3k-2) $$

Podemos separar la suma:
$$
\sum_{k=1}^{6}(3k-2) = 3\sum_{k=1}^{6} k - 2\sum_{k=1}^{6}1
$$
Sabemos que:
$$
\sum_{k=1}^{6} k = \frac{6(6+1)}{2} = 21 \quad \text{y} \quad \sum_{k=1}^{6}1 = 6
$$
Entonces:
$$
3 \times 21 - 2 \times 6 = 63 - 12 = 51
$$

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**d. Cálculo de** $$ \sum_{m=10}^{15} m^{2} $$

Calculamos cada término:
$$
\begin{aligned}
10^{2} &= 100 \
11^{2} &= 121 \
12^{2} &= 144 \
13^{2} &= 169 \
14^{2} &= 196 \
15^{2} &= 225 \
\end{aligned}
$$
Sumamos:
$$
100 + 121 = 221,\quad 221 + 144 = 365,\quad 365 + 169 = 534,\
534 + 196 = 730,\quad 730 + 225 = 955
$$
Por lo tanto, la suma es $$955$$.

---

**e. Cálculo de** $$ \sum_{i=1}^{6}(-1)^{i+1} i $$

Evaluamos cada término:
$$
\begin{aligned}
i=1:\quad &(-1)^{1+1}\cdot 1 = (-1)^2\cdot 1 = 1\cdot 1 = 1\[6mm]
i=2:\quad &(-1)^{2+1}\cdot 2 = (-1)^3\cdot 2 = -1\cdot 2 = -2\[6mm]
i=3:\quad &(-1)^{3+1}\cdot 3 = (-1)^4\cdot 3 = 1\cdot 3 = 3\[6mm]
i=4:\quad &(-1)^{4+1}\cdot 4 = (-1)^5\cdot 4 = -1\cdot 4 = -4\[6mm]
i=5:\quad &(-1)^{5+1}\cdot 5 = (-1)^6\cdot 5 = 1\cdot 5 = 5\[6mm]
i=6:\quad &(-1)^{6+1}\cdot 6 = (-1)^7\cdot 6 = -1\cdot 6 = -6\
\end{aligned}
$$
Sumamos:
$$
1 - 2 + 3 - 4 + 5 - 6 = -3
$$

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**f. Cálculo de** $$ \sum_{j=1}^{6}(-1)^{j} j $$

Evaluamos cada término:
$$
\begin{aligned}
j=1:\quad &(-1)^1 \cdot 1 = -1\[6mm]
j=2:\quad &(-1)^2 \cdot 2 = 1\cdot 2 = 2\[6mm]
j=3:\quad &(-1)^3 \cdot 3 = -1\cdot 3 = -3\[6mm]
j=4:\quad &(-1)^4 \cdot 4 = 1\cdot 4 = 4\[6mm]
j=5:\quad &(-1)^5 \cdot 5 = -1\cdot 5 = -5\[6mm]
j=6:\quad &(-1)^6 \cdot 6 = 1\cdot 6 = 6\
\end{aligned}
$$
Sumamos:
$$
-1 + 2 - 3 + 4 - 5 + 6 = 3
$$
**a.** 90 **b.** 12,207,031 **c.** 51 **d.** 955 **e.** -3 **f.** 3

Calcula las siguientes sumas.
a.
b.
c.
d.
e.
f.

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