Question

Write the series in the sigma notation. NO LINKS!!​

Answer 1

See below for answers and explanations

Explanation:

Problem 44:

Because there are 5 terms in the sequence, with each term getting multiplied by 3 having the first term be 3, this equates to the sigma notation
`\[ \sum_(n=1)^(5) 3^(n)\]`.

Problem 45:

Because there are 5 terms in the sequence, with each term getting multiplied by 1/2 having the first term be 80, this equates to the sigma notation
`\[ \sum_(n=1)^(5) 80((1)/(2))^(n-1)\]`.

Answer 2

9514 1404 393

Answer:

44. sum[n=1..5](3^n)

45. sum[n=0..3](80·2^-n)

Explanation:

The summation symbol is used to collapse all of the plus signs. The argument of the summation symbol is the general term of the series. These are both geometric series, so the general term can be written in the form ...

an = a1·r^(n-1)

where a1 is the first term and r is the common ratio.

The "-1" can be removed from the exponent if the summation starts with n=0 instead of n=1. Sometimes the multiplication by a1 can be accomplished by adjusting the exponent of the exponential term.

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49. The first term is 3 and the common ratio is 3, so the general term is ...

an = 3·3^(n-1) = 3^n . . . for n starting at 1

There are 5 terms in this series, so it can be written as ...

`\displaystyle S=\sum_(n=1)^5{3^n}`

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50. The first term is 80 and the common ratio is 1/2, or 2^-1. Then the general term is ...

an = 80·(2^-1)^(n-1) . . . for n starting at 1

If we let start at 0, then the general term is ...

an = 80·2^-n

There are 4 terms in the series, so it can be written as ...

`\displaystyle S=\sum_(n=0)^3{(80\cdot2^(-n))}`