How do you use summation notation to expression the sum # 5+15+45+...+3645#?

Question

Ezra Adams · Accepted Answer

# 5+15+45+...+3645 = sum_(r=0)^6 \ 3^r*5= 5 \ sum_(r=0)^6 \ 3^r#

Let the given summation be denoted by #S#

Then:

# S = 5+15+45+...+3645# # \ \ \= 5+3*5+9*5+...+729*5# # \ \ \= 3^0*5+3^1*5+3^2*5+...+3^6*5# # \ \ \= sum_(r=0)^6 \ 3^r*5# # \ \ \= 5 \ sum_(r=0)^6 \ 3^r#

Axel Alvarez · Answer

undefined To express the sum $5+15+45+...+3645$ using summation notation, you can use the formula for the general term of the sequence and the summation symbol. The sequence appears to be a geometric sequence where each term is obtained by multiplying the previous term by 3 and adding 5.

The general term of the sequence can be represented as $a_n = 5 \times 3^{n-1}$, where $n$ is the position of the term in the sequence.

Thus, the sum can be expressed as:

$$
\sum_{n=1}^{N} 5 \times 3^{n-1}
$$

where $N$ is the number of terms in the sequence. To find $N$, you can use the formula for the nth term:

$$a_n = 5 \times 3^{n-1}$$

Setting $a_n = 3645$ and solving for $n$:

$$3645 = 5 \times 3^{n-1}$$

$$3^{n-1} = \frac{3645}{5}$$

$$3^{n-1} = 729$$

$$n-1 = \log_3(729)$$

$$n-1 = 6$$

$$n = 7$$

Thus, there are 7 terms in the sequence. Therefore, the summation notation for the given sequence is:

$$
\sum_{n=1}^{7} 5 \times 3^{n-1}
$$

Elijah Abram · Answer

undefined The given series is a geometric series where each term is multiplied by 3 to get the next term, starting from 5. This can be expressed as $5, 15, 45, \ldots, 3645$, where the first term $a = 5$ and the common ratio $r = 3$.

The sum of a geometric series can be expressed using the formula:

$$ S_n = \frac{a(r^n - 1)}{r - 1} $$

To find $n$, the number of terms, we use the last term formula for a geometric series:

$$ a \cdot r^{n-1} = 3645 $$

Substituting $a = 5$ and $r = 3$:

$$ 5 \cdot 3^{n-1} = 3645 $$

$$ 3^{n-1} = \frac{3645}{5} $$

$$ 3^{n-1} = 729 $$

Since $729 = 3^6$, we have:

$$ 3^{n-1} = 3^6 $$

$$ n-1 = 6 $$

$$ n = 7 $$

Therefore, the sum of the series is the sum of the first 7 terms of this geometric sequence, which can be expressed in summation notation as:

$$ S = \sum_{k=0}^{6} 5 \cdot 3^k $$