If 2¹⁰ + 2⁹.3¹ + 2⁸ .3² +.....+ 2.3⁹ + 3¹⁰ = S - 2¹¹, then S is equal to :

Solution

Maths sir · 3 steps

Every term is 2ᵃ 3ᵇ with a + b = 10, so this is the factor of 3¹¹ - 2¹¹ in disguise.

1. Spot the pattern. The powers of 2 fall from 10 to 0 while the powers of 3 rise from 0 to 10. That gives 11 terms, each one 3/2 times the one before it. So the sum is a GP with first term 2¹⁰ and ratio 3/2.

   2¹⁰·((3/2)¹¹-1)/(3/2-1) = 2¹¹(3¹¹/2¹¹-1) = 3¹¹-2¹¹

2. Same via identity. This is the identity a¹¹-b¹¹ = (a-b)(a¹⁰+a⁹b+…+b¹⁰) with a = 3 and b = 2. Since a - b = 1, the bracket equals 3¹¹-2¹¹.
3. So. S - 2¹¹ = 3¹¹ - 2¹¹. Add 2¹¹ to both sides to get S = 3¹¹, which is option (d).

The idea to keep. A sum like aⁿ⁻¹ + aⁿ⁻²b + … + bⁿ⁻¹ is always (aⁿ - bⁿ)/(a - b). Spot it and the whole sum collapses in one line.

Answer from the NTA final answer key. Solution by PrepWiser.