Question

If S₁, S₂, and S₃ represent the number of n terms, 2n terms, and 3n of the first term of the geometry series, prove that:
A. S₁ (S₃ - S₂) = (S₂ - S₁) ²
B. S₁² + S₂² = S₁ (S₂ + S₃)

Answer 1

$\displaystyle S_1(S_3-S_2)=S_1\left(\frac{a(r^{3n}-1)}{r-1}-\frac{a(r^{2n}-1)}{r-1}\right)\\S_1(S_3-S_2)=S_1\left(\frac{a(r^{n}-1)(1+r^n+r^{2n})}{r-1}-\frac{a(r^{n}-1)(r^n+1)}{r-1}\right)\\S_1(S_3-S_2)=S_1\left(\frac{a(r^{n}-1)}{r-1}(1+r^n+r^{2n}-r^n-1)\right)\\S_1(S_3-S_2)=S_1\left(\frac{a(r^{n}-1)}{r-1}(r^{2n})\right)\\S_1(S_3-S_2)=\frac{a(r^{n}-1)}{r-1}\left(\frac{a(r^{3n}-r^{2n})}{r-1}\right)\\S_1(S_3-S_2)=\frac{a^2(r^{4n}-r^{3n}-r^{3n}+r^{2n})}{(r-1)^2}\\\displaystyle S_1(S_3-S_2)=\frac{a^2(r^{4n}-2r^{3n}+r^{2n})}{(r-1)^2}\\S_1(S_3-S_2)=\frac{a^2(r^{4n}-2r^{2n}\cdot r^n+r^{2n})}{(r-1)^2}\\S_1(S_3-S_2)=\frac{a^2(r^{2n}-r^n)^2}{(r-1)^2}\\S_1(S_3-S_2)=\frac{a^2(r^{2n}-1-(r^n+1))^2}{(r-1)^2}\\S_1(S_3-S_2)=\left(\frac{a(r^{2n}-1)-a(r^n+1)}{r-1}\right)^2\\S_1(S_3-S_2)=\left(\frac{a(r^{2n}-1)}{r-1}-\frac{a(r^{n}-1)}{r-1}\right)^2\\\boxed{\boxed{S_1(S_3-S_2)=\left(S_2-S_1\right)^2}}$

$\displaystyle S_1(S_2+S_3)=S_1\left(\frac{a(r^{2n}-1)}{r-1}+\frac{a(r^{3n}-1)}{r-1}\right)\\S_1(S_2+S_3)=\frac{a(r^{n}-1)}{r-1}\left(\frac{a(r^{3n}+r^{2n}-2)}{r-1}\right)\\S_1(S_2+S_3)=\frac{a^2(r^{4n}+r^{3n}-2r^n-r^{3n}-r^{2n}+2)}{(r-1)^2}\\S_1(S_2+S_3)=\frac{a^2(r^{4n}-r^{2n}-2r^n+2)}{(r-1)^2}\\S_1(S_2+S_3)=\frac{a^2(r^{4n}-2r^{2n}+1+r^{2n}-2r^n+1)}{(r-1)^2}\\S_1(S_2+S_3)=\frac{a^2((r^{2n}-1)^2+(r^{n}-1)^2)}{(r-1)^2}\\S_1(S_2+S_3)=\frac{a^2(r^{2n}-1)^2+a^2(r^{n}-1)^2}{(r-1)^2}\\\displaystyle S_1(S_2+S_3)=\frac{a^2(r^{2n}-1)^2}{(r-1)^2}+\frac{a^2(r^{n}-1)^2}{(r-1)^2}\\\boxed{\boxed{S_1(S_2+S_3)=S_2^2+S_1^2}}$

Answer 2

Let a is the first term and r is the common ratio of GP .also S1 , S2 and S3 are the number of n terms , 2n terms and 3n terms of the first terms of the GP .
so,
S1 = a.(rⁿ -1)/(r -1)
S2 = a.(r²ⁿ-1)/(r -1)
S3 = a.(r³ⁿ-1)/(r -1)

A) LHS = S1(S3 - S2)
= a.(rⁿ-1)/(r-1) { a.(r³ⁿ-1)-a.(r²ⁿ-1)}/(r -1)
=a²(rⁿ-1)/(r-1)² { r³ⁿ -1 - r²ⁿ+1 }
= a².(rⁿ-1)/(r-1)².r²ⁿ(rⁿ-1)
= a².(rⁿ-1)².r²ⁿ/(r-1)²
=a².(rⁿ.rⁿ -1.rⁿ)²/(r-1)²
= a².(r²ⁿ-rⁿ)²/(r -1)²
= a²{(r²ⁿ-1) -(rⁿ-1)}/(r -1)²
= { a.(r²ⁿ-1)/(r -1) - a.(rⁿ-1)/(r -1)}²
= {S2 - S1 }² = RHS

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B) LHS = S1² + S2²
= {a.(rⁿ-1)/(r-1)}² + {a.(r²ⁿ-1)/(r -1)}²
= a²/(r-1)² { (rⁿ-1)² + (r²ⁿ-1)²}
=a²/(r -1)²{ r²ⁿ+1 -2rⁿ + r⁴ⁿ +1 -2r²ⁿ}
= a²/(r -1)² { 2 - 2rⁿ + r⁴ⁿ -r²ⁿ}
= a²/(r -1)²{ r²ⁿ( rⁿ -1)(rⁿ+1) -2(rⁿ -1) }
= a².(rⁿ -1)/(r -1)²{ r³ⁿ + r²ⁿ -2 }
= {a².(rⁿ-1)/(r -1)} { (r³ⁿ -1)/(r -1) + (r²ⁿ-1)/(r -1)}
= { a.(rⁿ-1)/(r -1)} { a.(r³ⁿ-1)/(r -1) + a.(r²ⁿ -1)/(r -1)}
= S1 ( S3 + S2) = RHS