Example 15

Transcript

Example 15 Find the term independent of x in the expansion of ﷐﷐﷐3﷮𝑥﷯ + ﷐1﷮2 ﷐3﷮𝑥﷯﷯﷯﷮18﷯, x > 0. Step1 : Calculate general term of expansion ﷐﷐﷐3﷮𝑥﷯ + ﷐1﷮2 ﷐3﷮𝑥﷯﷯﷯﷮18﷯ We know that general term of expansion (a + b)n is Tr+1 = nCr (a)n–r . (a)n For general term of expansion ﷐﷐﷐3﷮𝑥﷯ + ﷐1﷮2 ﷐3﷮𝑥﷯﷯﷯﷮18﷯ Putting n = 18 , a = ﷐3﷮𝑥﷯ , b = ﷐1﷮2 ﷐3﷮𝑥﷯﷯ ∴ Tr+1 = 18Cr (﷐3﷮𝑥﷯)18 – r ﷐﷐﷐1﷮2 ﷐3﷮𝑥﷯﷯﷯﷮𝑟﷯ = 18Cr (﷐𝑥)﷮﷐1﷮3﷯ × (18−𝑟)﷯ ﷐﷐﷐1﷮2﷯× ﷐1﷮﷐3﷮𝑥﷯﷯﷯﷮𝑟﷯ = 18Cr (﷐𝑥)﷮﷐18 − 𝑟﷮3﷯ ﷯ ﷐﷐﷐1﷮2﷯﷯﷮𝑟﷯﷐﷐﷐1﷮﷐3﷮𝑥﷯﷯﷯﷮𝑟﷯ = 18Cr (﷐𝑥)﷮﷐18 − 𝑟﷮3﷯ ﷯ ﷐1﷮﷐2﷮𝑟﷯﷯﷐﷐﷐1﷮﷐𝑥﷮﷐1﷮3﷯﷯﷯﷯﷮𝑟﷯ = 18Cr (﷐𝑥)﷮﷐18 − 𝑟﷮3﷯ ﷯ ﷐1﷮﷐2﷮𝑟﷯﷯ ﷐1﷮﷐𝑥﷮﷐𝑟﷮3﷯﷯﷯ = 18Cr (﷐𝑥)﷮﷐18 − 𝑟﷮3﷯ ﷯ ﷐1﷮﷐2﷮𝑟﷯﷯ ﷐𝑥﷮﷐−𝑟﷮3﷯﷯ = 18Cr (﷐𝑥)﷮﷐18 − 𝑟﷮3﷯ ﷯ ﷐𝑥﷮﷐−𝑟﷮3﷯﷯ ﷐1﷮﷐2﷮𝑟﷯﷯ = 18Cr (﷐𝑥)﷮﷐18 − 𝑟﷮3﷯ − ﷐𝑟﷮3﷯﷯ ﷐1﷮﷐2﷮𝑟﷯﷯ = 18Cr (﷐𝑥)﷮﷐18 − 𝑟 − 𝑟﷮3﷯ ﷯ ﷐1﷮﷐2﷮𝑟﷯﷯ = 18Cr (﷐𝑥)﷮﷐18 − 2𝑟﷮3﷯ ﷯ ﷐1﷮﷐2﷮𝑟﷯﷯ ∴ Tr+1 = 18Cr (﷐𝑥)﷮﷐18 − 2𝑟﷮3﷯ ﷯ ﷐1﷮﷐2﷮𝑟﷯﷯ Step 2 We need to find the term independent of x So, power of x is 0 ﷐𝑥﷮﷐18 −2𝑟﷮3﷯﷯ = x0 Comparing power ﷐18 − 2𝑟﷮3﷯ = 0 18 – 2r = 0 18 = 2r ﷐18﷮2﷯ = r 9 = r r = 9 Putting r = 9 in (1) Tr+1 = 18Cr (﷐𝑥)﷮﷐18 − 2𝑟﷮3﷯ ﷯ ﷐1﷮﷐2﷮𝑟﷯﷯ T9+1 = 18C9 .﷐𝑥﷮﷐18 −3(9)﷮3﷯﷯. ﷐1﷮29﷯ = 18C9 .x0. ﷐1﷮29﷯ = 18C9 . ﷐1﷮29﷯ Hence, the term which is independent of x is T10 = 18C9 . ﷐1﷮29﷯