Q. 46 Find the sum or quantity without... [FREE SOLUTION]

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Chapter 4: Q. 46 (page 326)

Find the sum or quantity without completely expanding or calculating any sums.
Given ${鈭�}_{k = 0}^{25} k = 325$ and ${鈭�}_{k = 3}^{28} {(k - 3)}^{2} = 14, 910$ , find the value of ${鈭�}_{k = 3}^{25} (k^{2} - 5 k + 9)$ .

Short Answer

Expert verified

The sum is $13, 502$

Step by step solution

Step 1. Given information

${鈭�}_{k = 0}^{25} k = 325$

${鈭�}_{k = 3}^{28} {(k - 3)}^{2} = 14, 910$

Step 2. Rewrite the sum:

${鈭�}_{k = 3}^{25} (k^{2} - 5 k + 9) = {鈭�}_{k = 3}^{25} (k^{2} - 6 k + k + 9) = {鈭�}_{k = 3}^{25} (k^{2} - 6 k + 9 + k) = {鈭�}_{k = 3}^{25} ({(k - 3)}^{2} + k) = {鈭�}_{k = 3}^{25} {(k - 3)}^{2} + {鈭� k}_{k = 3}^{25}$

Step 3. To find ∑k=325k

${鈭�}_{k = 0}^{25} k = 325 0 + 1 + 2 + {鈭�}_{k = 3}^{25} k = 325 3 + {鈭� k}_{k = 3}^{25} = 325 {鈭� k}_{k = 3}^{25} = 325 - 3 {鈭� k}_{k = 3}^{25} = 322$

Step 4. To find ∑k=325k-32

${鈭�}_{k = 3}^{28} {(k - 3)}^{2} = 14, 910 {鈭�}_{k = 3}^{25} {(k - 3)}^{2} + {(26 - 3)}^{2} + {(27 - 3)}^{2} + {(28 - 3)}^{2} = 14, 910 {鈭�}_{k = 3}^{25} {(k - 3)}^{2} + {(23)}^{2} + {(24)}^{2} + {(25)}^{2} = 14, 910 {鈭�}_{k = 3}^{25} {(k - 3)}^{2} + 529 + 576 + 625 = 14, 910 {鈭�}_{k = 3}^{25} {(k - 3)}^{2} + 1730 = 14, 910 {鈭�}_{k = 3}^{25} {(k - 3)}^{2} = 14, 910 - 1730 {鈭�}_{k = 3}^{25} {(k - 3)}^{2} = 13, 180$

Step 5. Substitute the value of the sum obtained in steps 3 and 4 into step 2.

${鈭�}_{k = 3}^{25} (k^{2} - 5 k + 9) = {鈭�}_{k = 3}^{25} {(k - 3)}^{2} + {鈭�}_{k = 3}^{25} k = 13180 + 322 = 13502$

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91影视

Find the sum or quantity without completely expanding or calculating any sums.
Given ${鈭�}_{k = 0}^{25} k = 325$ and ${鈭�}_{k = 3}^{28} {(k - 3)}^{2} = 14, 910$ , find the value of ${鈭�}_{k = 3}^{25} (k^{2} - 5 k + 9)$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Rewrite the sum:

Step 3. To find ∑k=325k

Step 4. To find ∑k=325k-32

Step 5. Substitute the value of the sum obtained in steps 3 and 4 into step 2.

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91影视

Find the sum or quantity without completely expanding or calculating any sums.Given鈭�k=025k=325and鈭�k=328k-32=14,910, find the value of鈭�k=325k2-5k+9.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Rewrite the sum:

Step 3. To find ∑k=325k

Step 4. To find ∑k=325k-32

Step 5. Substitute the value of the sum obtained in steps 3 and 4 into step 2.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Logic and Functions

Calculus

Decision Maths

Probability and Statistics

Geometry

Study anywhere. Anytime. Across all devices.

Find the sum or quantity without completely expanding or calculating any sums.
Given ${鈭�}_{k = 0}^{25} k = 325$ and ${鈭�}_{k = 3}^{28} {(k - 3)}^{2} = 14, 910$ , find the value of ${鈭�}_{k = 3}^{25} (k^{2} - 5 k + 9)$ .