Q. 41

Question

Write each expression in Exercises 41 in one sigma notation (with some extra terms added to or subtracted from the sum, as necessary).

$2 \sum_{k = 0}^{100} a_{k} - \sum_{k = 3}^{101} a_{k}$

Step-by-Step Solution

Verified

Answer

The expression in one sigma is $2 a_{0} + 2 a_{1} + 2 a_{2} - a_{101} + \sum_{k = 3}^{100} a_{k}$ .

1Step 1: Given information

The given summation is $2 \sum_{k = 0}^{100} a_{k} - \sum_{k = 3}^{101} a_{k}$ .

2Step 2: Determine the expression is one sigma.

Rewrite the given summation in one sigma form.

$2 \sum_{k = 0}^{100} a_{k} - \sum_{k = 3}^{101} a_{k} = 2 (a_{0} + a_{1} + a_{2} + \sum_{k = 3}^{100} a_{k}) - \sum_{k = 3}^{101} a_{k} = 2 a_{0} + 2 a_{1} + 2 a_{2} + 2 \sum_{k = 3}^{100} a_{k} - (\sum_{k = 3}^{100} a_{k} + a_{101}) = 2 a_{0} + 2 a_{1} + 2 a_{2} + \sum_{k = 3}^{100} a_{k} - a_{101} = 2 a_{0} + 2 a_{1} + 2 a_{2} - a_{101} + \sum_{k = 3}^{100} a_{k}$

3Step 3: Write the conclusion

The one sigma notation is $2 a_{0} + 2 a_{1} + 2 a_{2} - a_{101} + \sum_{k = 3}^{100} a_{k}$ .

Q. 40

Q. 42

Other exercises in this chapter

Q. 39

Find a formula for each of the sums in Exercises 39, and then use these formulas to calculate each sum for n=100,500 and 1000∑k=1nk3-1n4

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Q. 40

Find a formula for each of the sums in Exercises 40, and then use these formulas to calculate each sum for n=100,500 and 1000

View solution

Q. 42

Write each expression in Exercises 41–43 in one sigma notation (with some extra terms added to or subtracted from the sum, as necessary). ∑k=14

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Q. 44

Find the sum or quantity without completely expanding or calculating any sums. Given ∑k=310ak=12 and ∑k=210ak=23, find a2.

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