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2+6+10+dots+(4n-2)=2n^(2)

2+6+10+…+(4n−2)=2n2 2+6+10+\ldots+(4 n-2)=2 n^{2}

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Full solution

Q. 2+6+10+…+(4n−2)=2n2 2+6+10+\ldots+(4 n-2)=2 n^{2}
  1. Identify first term and difference: Identify the first term (a1a_1) and the common difference (dd) of the arithmetic series.\newlinea1=2a_1 = 2, d=4d = 4
  2. Find nth term: Find the nth term ana_n using the formula an=a1+(n−1)da_n = a_1 + (n - 1)d.an=2+(n−1)4a_n = 2 + (n - 1)4an=2+4n−4a_n = 2 + 4n - 4an=4n−2a_n = 4n - 2
  3. Use sum formula: Use the sum formula for an arithmetic series: S=n2×(a1+an)S = \frac{n}{2} \times (a_1 + a_n).S=n2×(2+(4n−2))S = \frac{n}{2} \times (2 + (4n - 2))
  4. Simplify expression: Simplify the expression inside the parentheses.\newlineS=n2×(4n)S = \frac{n}{2} \times (4n)

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