Chapter 4: Q. 46 (page 326)

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

Short Answer

Expert verified

The sum is $13, 502$

Step by step solution

Step 1. Given information

$\sum_{k = 0}^{25} k = 325$

$\sum_{k = 3}^{28} {(k - 3)}^{2} = 14, 910$

Step 2. Rewrite the sum:

$\sum_{k = 3}^{25} (k^{2} - 5 k + 9) = \sum_{k = 3}^{25} (k^{2} - 6 k + k + 9) = \sum_{k = 3}^{25} (k^{2} - 6 k + 9 + k) = \sum_{k = 3}^{25} ({(k - 3)}^{2} + k) = \sum_{k = 3}^{25} {(k - 3)}^{2} + {\sum k}_{k = 3}^{25}$

Step 3. To find ∑k=325k

$\sum_{k = 0}^{25} k = 325 0 + 1 + 2 + \sum_{k = 3}^{25} k = 325 3 + {\sum k}_{k = 3}^{25} = 325 {\sum k}_{k = 3}^{25} = 325 - 3 {\sum k}_{k = 3}^{25} = 322$

Step 4. To find ∑k=325k-32

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

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

$\sum_{k = 3}^{25} (k^{2} - 5 k + 9) = \sum_{k = 3}^{25} {(k - 3)}^{2} + \sum_{k = 3}^{25} k = 13180 + 322 = 13502$

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

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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Find the sum or quantity without completely expanding or calculating any sums.Given∑k=025k=325and∑k=328k-32=14,910, find the value of∑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

Decision Maths

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Study anywhere. Anytime. Across all devices.

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