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Find the sum up to n terms of the series: 1.2.3 + 2.3.4 + … + n(n+1)(n+2). In this 1.2.3 represent the first term and 2.3.4 represent the second term.Let’s see an example to understand the concept better, Input: n = 5 Output: 420Explanation1.2.3 + 2.3.4 + 3.4.5 + 4.5.6 + 5.6.7 = 6 + 24 + 60 + 120 + 210 = 420nth term = n(n+1)(n+2); where n = 1, 2, 3, …= n(n^2+3n+2)=n^3 +3n^2 +2nNow, noteSum =n(n+1)/2 ; if nth term =n=n(n+1)(2n+1)/6 ; if nth term =n^2=n^2(n+1)^2/4 ; if nth term =n^3Hence the required sum =n^2(n+1)^2 /4 + ... Read More

An inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "square is inscribed in triangle" means precisely the same thing as "triangle is circumscribed about square".Biggest Square that can be inscribed within an Equilateral triangle −Biggest Square that can be inscribed within an Equilateral triangle −Let’s take an example, Input: 5 Output: 2.32ExplanationThe side of the square be x.Now, AH is perpendicular to DE.DE is parallel to BC, angle AED = angle ACB = 60In triangle EFC,    ⇒ Sin60 = x/ EC   ⇒ √3 / ... Read More

A Reuleaux triangle is a shape formed from the intersection of three circular disks, each having its center on the boundary of the other two. Its boundary is a curve of constant width, the simplest and best known such curve other than the circle itself. Constant width means that the separation of every two parallel supporting lines is the same, independent of their orientation. Because all its diameters are the same.The boundary of a Reuleaux triangle is a constant width curve based on an equilateral triangle. All points on a side are equidistant from the opposite vertex.To construct a Reuleaux ... Read More

A Reuleaux triangle is a shape formed from the intersection of three circular disks, each having its center on the boundary of the other two. Its boundary is a curve of constant width, the simplest and best known such curve other than the circle itself. Constant width means that the separation of every two parallel supporting lines is the same, independent of their orientation. Because all its diameters are the same.The boundary of a Reuleaux triangle is a constant width curve based on an equilateral triangle. All points on a side are equidistant from the opposite vertex.To construct a Reuleaux triangleFormula for Reuleaux triangleArea of the Reuleaux ... Read More

A Reuleaux triangle is a shape formed from the intersection of three circular disks, each having its center on the boundary of the other two. Its boundary is a curve of constant width, the simplest and best known such curve other than the circle itself. Constant width means that the separation of every two parallel supporting lines is the same, independent of their orientation. Because all its diameters are the same.The boundary of a Reuleaux triangle is a constant width curve based on an equilateral triangle. All points on a side are equidistant from the opposite vertex.To construct a Reuleaux ... Read More

A Reuleaux triangle is a shape formed from the intersection of three circular disks, each having its center on the boundary of the other two. Its boundary is a curve of constant width, the simplest and best known such curve other than the circle itself. Constant width means that the separation of every two parallel supporting lines is the same, independent of their orientation. Because all its diameters are the same.The boundary of a Reuleaux triangle is a constant width curve based on an equilateral triangle. All points on a side are equidistant from the opposite vertex.To construct a Reuleaux ... Read More

A Reuleaux triangle is a shape formed from the intersection of three circular disks, each having its center on the boundary of the other two. Its boundary is a curve of constant width, the simplest and best known such curve other than the circle itself. Constant width means that the separation of every two parallel supporting lines is the same, independent of their orientation. Because all its diameters are the same.The boundary of a Reuleaux triangle is a constant width curve based on an equilateral triangle. All points on a side are equidistant from the opposite vertex.To construct a Reuleaux ... Read More

A Reuleaux triangle is a shape formed by using intersection of three circle, in such a way that each having its center on the boundary of the other two circles. Its boundary is a curve of constant width, the simplest and best known such curve other than the circle itself. Constant width means that the separation of every two parallel supporting lines is the same, independent of their orientation. Because all its diameters are the same.The boundary of a Reuleaux triangle is a constant width curve based on an equilateral triangle. All points on a side are equidistant from the ... Read More

Betrothed numbers are pair of two numbers in a way that the sum of their divisors when added by is equal to another number.For example (a,  b) are a pair of betrothed numbers if s(a) = b + 1 and s(b) = a + 1, where s(b) is the aliquot sum of b: an equivalent condition is that σ(a) = σ(b) = a + b + 1, where σ denotes the sum-of-divisors function.The first few pairs of betrothed numbers are: (48,  75), (140,  195), (1050,  1925), (1575,  1648), (2024,  2295), (5775,  6128).All known pairs of betrothed numbers have opposite parity. Any pair of the same parity must exceed 1010.AlgorithmStep 1: Find the sum of all divisors for both numbers. Step 2: Finally ... Read More

Array is a container of elements of Same data types length needs to be defined beforehand. And an element can appear in any order and any number of times in an array. so in this program we will find elements that appear more than once in an array.Problem description − We have given an array arr[] in which we have to find which of the element are repeating in the array an app to print them.Let’s take an example to understand this better.Example, Input: arr[] = {5, 11, 11, 2, 1, 4, 2} Output: 11 2ExplanationWe have an array arr ... Read More