Video: Evaluating the Sum of a Finite Series after Expanding It

Expand and then evaluate βˆ‘_(π‘Ÿ = 1)^(4) (2^(π‘Ÿ) βˆ’ 52).

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Video Transcript

Expand and then evaluate the sum between π‘Ÿ equals one and π‘Ÿ equals four of two to the power π‘Ÿ minus 52.

Our first step is to substitute each of the integer values between one and four into the expression. When π‘Ÿ is equal to one, we have two to the power of one minus 52. When π‘Ÿ is equal to two, we have two squared minus 52. When π‘Ÿ is equal to three, we have two cubed minus 52. And finally, when π‘Ÿ is equal to four, we have two to the power of four minus 52.

Our next step is to evaluate each of the four terms and then find their sum. Two to the power of one is equal to two. And two minus 52 gives us negative 50. Two squared is equal to four. Subtracting 52 from this gives us negative 48. Two cubed is equal to eight. And eight minus 52 equals negative 44. Finally, two to the power of four is equal to 16. And subtracting 52 from this gives us negative 36.

As we’re adding negative numbers, this can be rewritten as negative 50 minus 48 minus 44 minus 36. This is equal to negative 178. The value of the sum of two to the power π‘Ÿ minus 52 for values of π‘Ÿ between one and four is negative 178.

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