Question 14 and 15, Exercise 4.2

Solutions of Question 14 and 15 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

Find '$b$' if $10$ is A.M between $b$ and $20$.

Solution.

Let $a= b$ and $b=20$. Then $$\begin{align*} &\text{A.M.} = \frac{a + b}{2} \\ \implies & 10 = \frac{b + 20}{2} \\ \implies & 20 = b + 20 \\ \implies & b = 20 - 20 \\ \implies & b = 0 \end{align*}$$ Hence $b = 0$.

Find $x$ and $y$ if $2$ and $13$ are two arithmetic means between $x$ and $y$.

Solution.

Given: $2$ and $13$ are two arithmetic means between $x$ and $y$.
This gives $x$, $2$, $13$, $y$ are in A.P.

Here $a_1 = x$ and $d=13-2=11$.

Now $$\begin{align*} & a_2= a_1+d\\ \implies &2 = x + 11 \\ \implies & \boxed{x=-9} \end{align*}$$ Also $$\begin{align*} &a_4=a_1+3d\\ \implies &y=x+3(11)\\ \implies &y=-9 + 33 \\ \implies &\boxed{y=24}\end{align*}$$ Hence $x=-9$ and $y=24$.

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