Ex 8.2

Chapter 8 Class 11 Sequences and Series
Serial order wise

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### Transcript

Ex9.3, 27 Find the value of n so that (π^(π + 1) +π^(π + 1))/(π^(π ) +π^π ) may be the geometric mean between a and b. We know that geometric mean between a & b is a & b = βab It is given that G.M. between a & b = (π^(π + 1) +π^(π + 1))/(π^(π ) +π^(π ) ) βab = (π^(π + 1) +π^(π + 1))/(π^(π ) +π^(π ) ) γ"(ab)" γ^(1/2) = (π^(π + 1) +π^(π + 1))/(π^(π ) +π^(π ) ) γ"(ab)" γ^(1/2) (an +bn) = an + 1 + bn + 1 γ"a" γ^(1/2) π^(1/2) (an +bn) = an + 1 + bn + 1 γ"a" γ^(1/2) an π^(1/2) + γ"a" γ^(1/2) bn π^(1/2) = an + 1 + bn + 1 π^(1/2 + π ) π^(1/2) + γ"a" γ^(1/2) π^(1/2 + π )= an + 1 + bn + 1 π^(1/2 + π ) π^(1/2) β an + 1 = bn + 1 β γ"a" γ^(1/2) π^(1/2 + π ) π^(1/2 + π ) π^(1/2) β π^(π + 1/2 + 1/2) = π^(π + 1/2 + 1/2) β π^(1/2) π^(1/2 + π ) π^(1/2 + π ) [π^(1/2) β π^(1/2)] = π^(π + 1/2 ) [π^(1/2) β π^(1/2)] π^(1/2 + π )= π^(π + (1 )/2 "[" π^(1/2) " β " π^(1/2) "] " )/(π^(1/2) " β " π^(1/2) ) π^(1/2 + π )= π^(π +1/2) (π/π)^(1/2 + π) = 1 (π/π)^(1/2 + π)= (π/π)^0 Comparing powers 1/2 + n = 0 n = β 1/2 Hence value of n is - 1/2