[tex]\sum_{n=0}^a 2^n(7-i)^{4 n}[/tex]
Answer (2)
Rewrite the summand: 2 n ( 7 − i ) 4 n = ( 2 ( 7 − i ) 4 ) n .
Recognize the sum as a geometric series with z = 2 ( 7 − i ) 4 .
Apply the geometric series formula: ∑ n = 0 a z n = 1 − z 1 − z a + 1 .
Simplify and substitute to get the final answer: 1 − ( 4216 − 2688 i ) 1 − ( 4216 − 2688 i ) a + 1 .
Explanation
Problem Analysis We are given the sum ∑ n = 0 a 2 n ( 7 − i ) 4 n . Our goal is to evaluate this sum and express it in a simplified form.
Rewriting the Summand Let's rewrite the summand to identify a pattern. We have 2 n ( 7 − i ) 4 n = 2 n (( 7 − i ) 4 ) n = ( 2 ( 7 − i ) 4 ) n . This allows us to recognize the sum as a geometric series.
Identifying Geometric Series Let z = 2 ( 7 − i ) 4 . Then the sum becomes ∑ n = 0 a z n . This is a geometric series with first term 1 , common ratio z , and a + 1 terms.
Applying Geometric Series Formula The formula for the sum of a geometric series is given by ∑ n = 0 a z n = 1 − z 1 − z a + 1 . We will use this formula to find a closed-form expression for the sum.
Simplifying (7-i)^4 Now we need to simplify ( 7 − i ) 4 . We can do this by first finding ( 7 − i ) 2 and then squaring the result: ( 7 − i ) 2 = ( 7 − i ) ( 7 − i ) = 49 − 7 i − 7 i + i 2 = 49 − 14 i − 1 = 48 − 14 i .
Then, we square this result: ( 7 − i ) 4 = ( 48 − 14 i ) 2 = ( 48 − 14 i ) ( 48 − 14 i ) = 4 8 2 − 2 ( 48 ) ( 14 i ) + ( 14 i ) 2 = 2304 − 1344 i − 196 = 2108 − 1344 i .
Calculating z Now we can find z = 2 ( 7 − i ) 4 = 2 ( 2108 − 1344 i ) = 4216 − 2688 i .
Final Summation Substitute z = 4216 − 2688 i back into the geometric series formula: n = 0 ∑ a 2 n ( 7 − i ) 4 n = 1 − ( 4216 − 2688 i ) 1 − ( 4216 − 2688 i ) a + 1 .
Final Answer Therefore, the final answer is 1 − ( 4216 − 2688 i ) 1 − ( 4216 − 2688 i ) a + 1 .
Examples
Geometric series appear in many areas of mathematics and physics. For example, they are used in calculating the present value of an annuity, analyzing the motion of a bouncing ball, and modeling population growth under certain conditions. Understanding how to sum a geometric series is a fundamental skill in these fields.
To solve the series ∑ n = 0 a 2 n ( 7 − i ) 4 n , we can rewrite it as a geometric series, apply the geometric series formula, and evaluate the complex exponentiation. The final answer is 1 − ( 4216 − 2688 i ) 1 − ( 4216 − 2688 i ) a + 1 . This simplifies the process of summing the series effectively.
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