A sequence of five real numbers forms an arithmetic progression. The sum of the first and fifth terms is 20, and the product of the second and fourth terms is 48. Find the third term.

["Title: Solving an Arithmetic Progression: Finding the Third Term from Key Conditions", "A sequence of five real numbers forms an arithmetic progression (AP) — a classic topic in algebra that highlights the beauty of uniform differences between consecutive terms. In this article, we explore a specific problem where mathematical properties lead to a clear and elegant solution, revealing the third term in the sequence using logical reasoning and basic formulas.", "---", "### Understanding Arithmetic Progression (AP)", "An arithmetic progression is a sequence where each term increases (or decreases) by a constant difference. Given five terms:", "[\na, \quad a + d, \quad a + 2d, \quad a + 3d, \quad a + 4d\n]", "Here,\n- (a) is the first term\n- (d) is the common difference", "The sequence is fully defined by these two parameters.", "---", "### Given Conditions", "From the problem, we have two key conditions:", "1. Sum of the first and fifth terms is 20:\n[\na + (a + 4d) = 20 \implies 2a + 4d = 20 \quad \ ext{(Equation 1)}\n]", "2. Product of the second and fourth terms is 48:\n[\n(a + d)(a + 3d) = 48 \quad \ ext{(Equation 2)}\n]", "Our goal is to find the third term, which is:\n[\na + 2d\n]", "---", "### Step 1: Simplify Equation 1", "From Equation 1:\n[\n2a + 4d = 20\n]\nDivide both sides by 2:\n[\na + 2d = 10 \quad \ ext{(Key insight!)}\n]", "Notice that (a + 2d) is exactly the third term of the AP! So we already have our answer — $10$ — but let’s verify consistency by checking Equation 2.", "---", "### Step 2: Express Equation 2 in Terms of Known Quantity", "We know:\n[\na + 2d = 10\n]\nLet’s use this to express the second and fourth terms:", "- Second term: (a + d = (a + 2d) - d = 10 - d)\n- Fourth term: (a + 3d = (a + 2d) + d = 10 + d)", "Now compute their product:", "[\n(10 - d)(10 + d) = 100 - d^2 = 48\n]", "Solve for (d^2):", "[\n100 - d^2 = 48 \implies d^2 = 52 \implies d = \pm \sqrt{52} = \pm 2\sqrt{13}\n]", "Even though (d) has real values, the third term (a + 2d = 10) remains unaffected — the value is uniquely determined from the first condition.", "---", "### Verification", "Let’s confirm with (d = 2\sqrt{13}):\nThen (a + 2d = 10) (consistent), so first term:\n[\na = 10 - 2d = 10 - 4\sqrt{13}\n]", "Second term:\n[\na + d = 10 - 3\sqrt{13}\n]\nFourth term:\n[\na + 3d = 10 + \sqrt{13}\n]\nProduct:\n[\n(10 - 3\sqrt{13})(10 + \sqrt{13}) = 100 + 10\sqrt{13} - 30\sqrt{13} - 3 \cdot 13 = 100 - 20\sqrt{13} - 39 = 61 - 20\sqrt{13}\n]", "Wait — this does not equal 48! What’s wrong?", "Ah — here’s the key: we assumed (a + 2d = 10) directly from Equation 1, which is correct. But in verification, we failed to ensure Equation 2 holds. That suggests we must solve both equations simultaneously — even though one gave a clean result.", "Let’s correct the approach.", "---", "### Correct Step: Use Equation 1 to Eliminate (a)", "From Equation 1:\n[\na = 10 - 2d\n]", "Substitute into Equation 2:", "[\n(a + d)(a + 3d) = (10 - 2d + d)(10 - 2d + 3d) = (10 - d)(10 + d) = 100 - d^2 = 48\n]", "[\n100 - d^2 = 48 \implies d^2 = 52 \implies d = \pm 2\sqrt{13}\n]", "Now compute the third term:\n[\na + 2d = (10 - 2d) + 2d = 10\n]", "The (d) cancels — confirming that regardless of the sign of (d), the third term is always 10.", "So even though individual terms change with (d), the symmetry in the progression and the given conditions force the middle term to be 10.", "---", "### Conclusion", "By combining algebraic simplification with verification, we confirm that:", "- The sum condition (a + (a+4d) = 20) leads directly to (a + 2d = 10).\n- The product condition constrains (d), but does not alter the third term.\n- Thus, the third term of the arithmetic progression is 10.", "This elegant result illustrates how AP symmetries simplify problems — even when intermediate steps involve square roots and verification is essential.", "If you're studying arithmetic sequences, remember: the average of the first and fifth terms is the third term. Since their sum is 20, their average is 10 — and that’s your answer.", "---", "Key Takeaways:\n- In an AP, the middle (third) term is the average of symmetric terms.\n- Using (a + (a+4d) = 20), the third term is ( \frac{20}{2} = 10 ).\n- The product condition ensures consistency but does not override this key property.", "Use this insight to solve similar problems with confidence!", "---", "Keywords: arithmetic progression, AP, third term, five-term AP, algebra problem, common difference, sum and product, root system, quadratic approach, middle term of AP, solve arithmetic sequence.", "Meta Description:\nDiscover how to find the third term in a five-number arithmetic progression using sum and product of outer terms. We prove that $ a + 2d = 10 $ directly from the given conditions, confirming the third term is always 10."]