Chapter 13: Q. 28 (page 1004)

Evaluate the sums in Exercises $23 - 28$
$\sum_{i = 1}^{l} \sum_{j = 1}^{m} \sum_{k = 1}^{m} i j^{2} k^{3}$

Short Answer

Expert verified

The value of summation is

$\frac{l m n^{2} (l + 1) (m + 1) (2 m + 1) (n + 1)^{2}}{48}$

Step by step solution

Step 1. Given information

An expression s given as $\sum_{i = 1}^{l} \sum_{j = 1}^{m} \sum_{k = 1}^{m} i j^{2} k^{3}$

Step 2. Evaluating sum

Separate the first to last as,

$\sum_{i = 1}^{l} \sum_{j = 1}^{m} \sum_{k = 1}^{k} i j^{2} k^{3} = \sum_{i = 1}^{l} \sum_{j = 1}^{m} (\sum_{i = 1}^{n} i j^{2} k^{3})$

Now simplify the summation as,

$\sum_{i = 1}^{n} i j^{2} k^{3} = i j^{2} 1^{3} + i j^{2} 2^{3} + i j^{2} 3^{3} + \dots + i j^{2} n^{3} = i j^{2} (1^{3} + 2^{3} + 3^{3} + \dots + n^{3}) = i j^{2} \frac{n^{2} (n + 1)^{2}}{4}$

Now simplify the each terms as above,

$\sum_{k = 1}^{1} \sum_{j = 1}^{m} \sum_{k = 1}^{n} i j^{2} k^{3} = \sum_{i = 1}^{1} \sum_{j = 1}^{m} (\sum_{k = 1}^{n} i j^{2} k^{3}) = \sum_{j = 1}^{1} (\sum_{j = 1}^{m} i j^{2} \frac{n^{2} (n + 1)^{2}}{4}) \sum_{j = 1}^{m} i^{2} \frac{n^{2} (n + 1)^{2}}{4} = i \frac{n^{2} (n + 1)^{2}}{4} (\sum_{j = 1}^{m} j^{2}) = i \frac{n^{2} (n + 1)^{2}}{4} (1^{2} + 2^{2} + 3^{2} + \dots + m^{2}) = i \times \frac{n^{2} (n + 1)^{2}}{4} \times \frac{m (m + 1) (2 m + 1)}{6} \sum_{i = 1}^{1} \sum_{j = 1}^{m} \sum_{k = 1}^{n} i j^{2} k^{3} = \sum_{i = 1}^{1} (\sum_{j = 1}^{m} i j^{2} \frac{n^{2} (n + 1)^{2}}{4}) = \sum_{i = 1}^{l} (i \times \frac{n^{2} (n + 1)^{2}}{4} \times \frac{m (m + 1) (2 m + 1)}{6}) = \frac{n^{2} (n + 1)^{2}}{4} \times \frac{m (m + 1) (2 m + 1)}{6} \times (\sum_{i = 1}^{1} (i)) = \frac{n^{2} (n + 1)^{2}}{4} \times \frac{m (m + 1) (2 m + 1)}{6} \times (1 + 2 + 3 + \dots + l) = \frac{n^{2} (n + 1)^{2}}{4} \times \frac{m (m + 1) (2 m + 1)}{6} \times \frac{l (l + 1)}{2} = \frac{l m n^{2} (l + 1) (m + 1) (2 m + 1) (n + 1)^{2}}{48}$

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Evaluate the sums in Exercises $23 - 28$
$\sum_{i = 1}^{l} \sum_{j = 1}^{m} \sum_{k = 1}^{m} i j^{2} k^{3}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Evaluating sum

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Evaluate the sums in Exercises 23–28∑i=1l∑j=1m∑k=1mij2k3

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Step by step solution

Step 1. Given information

Step 2. Evaluating sum

One App. One Place for Learning.

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Evaluate the sums in Exercises $23 - 28$
$\sum_{i = 1}^{l} \sum_{j = 1}^{m} \sum_{k = 1}^{m} i j^{2} k^{3}$