If the third and fourth terms in the expansion (2 x+ 1 8 )¹⁰ are equal, then the value of x is

Step-by-step solution

The general term in the expansion of (a + b)^ n is given by T_ r+1 = ^ n C_ r a^ n-r b^ r . For the expansion (2x + 1 8 )¹⁰ , the third term T₃ (where r=2 ) is: T₃ = ¹⁰C₂ (2x)¹⁰⁻² ( 1 8 )² = ¹⁰C₂ (2x)⁸ ( 1 8 )² The fourth term T₄ (where r=3 ) is: T₄ = ¹⁰C₃ (2x)¹⁰⁻³ ( 1 8 )³ = ¹⁰C₃ (2x)⁷ ( 1 8 )³ Given T₃ = T₄ , we have: ¹⁰C₂ (2x)⁸ ( 1 8 )² = ¹⁰C₃ (2x)⁷ ( 1 8 )³ Dividing both sides by (2x)⁷ ( 1 8 )² : ¹⁰C₂ (2x) = ¹⁰C₃ ( 1 8 ) Calculating the combinations: ¹⁰C₂ = 10 9 2 = 45 and ¹⁰C₃ = 10 9 8 3 2 1 = 120 . 45(2x) = 1