class 11

Math

Algebra

Sequences and Series

If m is the A.M. of two distinct real numbers $l$and $n(l,n>1)$and$G_{1},G_{2}$ and $G_{3}$ are three geometric means between $l$and n, then $G_{1}+2G_{2}+G_{3}$equals,

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In which of the following situations, does the list of numbers involved make an arithmetic progression, and why?(i) The taxi fare after each km when the fare is Rs 15 for the first km and Rs 8 for each additional km.(ii) The amount of air present in a cylinder when a vacuum pump removes 1/4 of the air remaining in the cylinder at a time. (iii) The cost of digging a well after every metre of digging, when it costs Rs.150 for the first metre and rises by Rs. 50 for each subsequent metre. (iv) The amount of money in the account every year, when \displaystyle{10000}{i}{s}{d}{e}{p}{o}{s}{i}{t}{e}{d}{a}{t}{c}{o}{m}{p}{o}{u}{n}{d}\int{e}{r}{e}{s}{t}{a}{t}{8}\%{p}{e}{r}{a}\cap{u}{m}.

A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.

Find the $20_{th}$and $n_{th}$terms of the G.P. $25 $,$45 $,$85 ,˙˙˙$

Find a G.P. for which sum of the first two terms is $−4$and the fifth term is 4 times the third term.

Fill in the blanks in the following table, given that a is the first term, d the common difference and $a_{n}$the nth term of the AP:

The $5_{th}$, $8_{th}$and $11_{th}$terms of a G.P. are p, q and s, respectively. Show that $q_{2}=ps$.

The $17_{th}$ term of an AP exceeds its $10_{th}$ term by 7. Find the common difference.

Find the value of n so that $a_{n}+b_{n}a_{n+1}+b_{n+1} $may be the geometric mean between a and b.