Question

IF A = 2^65, B=2^64+2^63+............+2^1+ 2^0and C=2^64+2^63+.....+2^2 +2^1 then prove that
2A =B+C+3...​

Answer 1

||✪✪ QUESTION ✪✪||

IF A = 2^65, B=2^64+2^63+............+2^1+ 2^0and C=2^64+2^63+.....+2^2 +2^1 then prove that

2A =B+C+3... ?

|| ✰✰ ANSWER ✰✰ ||

Given,

→ A = 2^65 ---------- Equation ❶

→ B = 2^64+2^63+............+2^1+ 2^0

Here, we can see That, B is in GP, with

→ First term (a) = 2^0

→ r = 2

→ n = 65

Using GP sum formula :-

➼ Sₙ = a(rⁿ - 1)/ (r - 1)

putting values we get,

→ Sₙ = 2^0 ( 2^65 - 1) /(2-1)

→ Sₙ = (2^65 - 1) (since a^0 = 1 )

→ B = (2^65 - 1) ------------------ Equation ❷

Similarly,

→ C = 2^64+2^63+.....+2^2 +2^1

→ C = 2^1(2^64 - 1) /(2 - 1)

→ C = 2(2^64 - 1)

→ C = ( 2^65 - 2 ) ---------------- Equation ❸

________________________

Now, we have to Prove ,

➻ 2A = B + C + 3

Putting values From Equation ❶, ❷ and ❸ , we get,

➻ 2(2^65) = (2^65 - 1) + (2^65 - 2) + 3

➻2*2^65 = 2^65 + 2^65 - 3 + 3

➻2^66 = 1*2^65 + 1*2^65

Using [p(a)^n + q(a)^n = (p + q)a^n] in RHS now,

➻2^66 = (1+1)(2^65)

➻ 2^66 = 2 * 2^65

➻ 2^66 = 2^66 ⟪☙☙ Hence, Proved ☙☙⟫

꧁____________________꧂

Answer 2

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$\huge\mathfrak\red{Answer:-}$

Refer to the attachment!!!

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