Between $$1$$ and $$31$$ are inserted M arithmetic means so that the ratio of the $$7th$$ and (M$$ $$-$$ $$1) th means is $$5$$:$$9$$, then the value of m is

Question

Between $$1$$ and  $$31$$  are  inserted  M arithmetic means so that the ratio of the $$7th$$  and  (M$$ $$-$$ $$1) th  means is $$5$$:$$9$$,  then the  value  of m is

Infinity Learn · Accepted Answer

let $$A1$$, $$A2$$,.,. , Am be ,m arithmetic means between $$1$$ and $$31$$. ⇒Then $$A1$$,$$A2$$,…,$$Am31$$ is an AP with common difference.$$d=31$$−$$1m+1=30m+1$$ ∵$$d=b$$−$$an+1$$∴  $$A7=1+7d=1+7$$×$$30m+1=m+211m+1and$$  Am−$$1=1+(m$$−$$1)d=1+30m+1(m$$−$$1)=31m$$−$$29m+1Now$$,  $$A7Am$$−$$1=59m+21131m$$−$$29=59$$⇒ $$9m+1899=155m$$−$$145$$⇒ $$146m=2044$$⇒ $$m=14$$ (given)

Between 1 and 31 are inserted M arithmetic means so that the ratio of the 7th and (M - 1) th means is 5:9, then the value of m is

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material