Fewpal
+4 votes
981 views
in Mathematics by (64.2k points)

If the first and the nth term of a G.P. are a and b, respectively, and if P is the product of n terms, prove that P2 = (ab)n.

1 Answer

+5 votes
by (13.5k points)
selected by
 
Best answer

Solution:

The first term of the G.P is a and the last term is b.

Therefore, the G.P. is aarar2ar3, … arn–1, where r is the common ratio.

b = arn–1 … (1)

P = Product of n terms

= (a) (ar) (ar2) … (arn–1)

= (a × a ×…a) (r × r2 × …rn–1)

an r 1 + 2 +…(n–1) … (2)

Here, 1, 2, …(n – 1) is an A.P.

∴1 + 2 + ……….+ (n – 1)

equals fraction numerator n minus 1 over denominator 2 end fraction open square brackets 2 plus left parenthesis n minus 1 minus 1 right parenthesis cross times 1 close square brackets equals fraction numerator n minus 1 over denominator 2 end fraction open square brackets 2 plus n minus 2 close square brackets equals fraction numerator n open parentheses n minus 1 close parentheses over denominator 2 end fractionP space equals space a to the power of n r to the power of fraction numerator n open parentheses n minus 1 close parentheses over denominator 2 end fraction end exponenttherefore space P squared space equals space a to the power of 2 n end exponent space r to the power of n open parentheses n minus 1 close parentheses end exponentspace space space space space space space space space space space space equals space open square brackets a squared space r to the power of left parenthesis n minus 1 right parenthesis end exponent close square brackets to the power of nspace space space space space space space space space space space space equals open square brackets a cross times a r to the power of n minus 1 end exponent close square brackets to the power of nspace space space space space space space space space space space space equals open parentheses a b close parentheses to the power of n space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets U sin g space left parenthesis 1 right parenthesis close square brackets

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.

Categories

...