$\boxed{\frac{27}{16}}$Question: Compute the value of $ \left( \sum_{k=1}^{6} k^2 \right) \times \left( \prod_{j=1}^{3} (2j + 1) \right) $.
["Unlock the Power of Fractions and Arithmetic: Compute $ \left( \sum_{k=1}^{6} k^2 \right) \ imes \left( \prod_{j=1}^{3} (2j + 1) \right) $", "When faced with values like $ \frac{27}{16} $, mathematics often leads us beyond simple fractions—into clean computations rooted in arithmetic, summation, and multiplication. In this article, we explore a compound expression involving a sum of squares and a product of odd sequence terms, arriving at a precise value.", "---", "### Step 1: Compute the Sum $ \sum_{k=1}^{6} k^2 $", "We start by evaluating:", "$$\n\sum_{k=1}^{6} k^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2\n$$", "Calculating each square:", "$$\n1 + 4 + 9 + 16 + 25 + 36\n$$", "Now sum step-by-step:", "- $ 1 + 4 = 5 $\n- $ 5 + 9 = 14 $\n- $ 14 + 16 = 30 $\n- $ 30 + 25 = 55 $\n- $ 55 + 36 = 91 $", "So,\n$$\n\sum_{k=1}^{6} k^2 = 91\n$$", "---", "### Step 2: Compute the Product $ \prod_{j=1}^{3} (2j + 1) $", "Now evaluate:", "$$\n\prod_{j=1}^{3} (2j + 1) = (2 \cdot 1 + 1)(2 \cdot 2 + 1)(2 \cdot 3 + 1)\n$$", "$$\n= (3)(5)(7)\n$$", "Multiplying:", "- $ 3 \ imes 5 = 15 $\n- $ 15 \ imes 7 = 105 $", "So,\n$$\n\prod_{j=1}^{3} (2j + 1) = 105\n$$", "---", "### Step 3: Multiply the Two Results", "Now compute the full expression:", "$$\n\left( \sum_{k=1}^{6} k^2 \right) \ imes \left( \prod_{j=1}^{3} (2j + 1) \right) = 91 \ imes 105\n$$", "Break down the multiplication using distributive property:", "$$\n91 \ imes 105 = 91 \ imes (100 + 5) = (91 \ imes 100) + (91 \ imes 5)\n$$", "- $ 91 \ imes 100 = 9100 $\n- $ 91 \ imes 5 = 455 $", "Add them:", "$$\n9100 + 455 = 9555\n$$", "---", "### Final Answer", "$$\n\boxed{9555}\n$$", "---", "This calculation shows how combining summation and products—values like $ \frac{27}{16} $—can lead to elegant results through careful arithmetic. Whether in education, programming, or engineering, such expressions demonstrate the beauty and utility of foundational math."]
