If the , and terms of a GP are , and respectively, prove that are in GP.

We will use the formula for the term of a Geometric Progression. Step 1 — Express terms Let the first term of the GP be . Let the common ratio of the GP be . The term of a GP is . The term is . > The term is . > The term is . > Step 2 — Check GP condition For to be in GP, the ratio must equal . Let's find the ratio . > Now, let's find the ratio . > We see that and . Therefore, . This means are in GP. Answer > (i) The terms are in GP.

Predicting What Comes Next: Sequences and Progressions | EOT

Question 12

If the $4^{\text{th}}$ , $10^{\text{th}}$ and $16^{\text{th}}$ terms of a GP are $x$ , $y$ and $z$ respectively, prove that $x, y, z$ are in GP.

Solution

We will use the formula for the $n^{\text{th}}$ term of a Geometric Progression.

Step 1 — Express terms

Let the first term of the GP be $a$ . Let the common ratio of the GP be $r$ . The $n^{\text{th}}$ term of a GP is $ar^{n-1}$ .

The $4^{\text{th}}$ term is $x$ . $x = ar^{4-1}$

$\boxed{x = ar^3}$

The $10^{\text{th}}$ term is $y$ . $y = ar^{10-1}$

$\boxed{y = ar^9}$

The $16^{\text{th}}$ term is $z$ . $z = ar^{16-1}$

$\boxed{z = ar^{15}}$

Step 2 — Check GP condition

For $x, y, z$ to be in GP, the ratio $y/x$ must equal $z/y$ . Let's find the ratio $y/x$ . $y/x = \frac{ar^9}{ar^3}$ $y/x = r^{9-3}$

$\boxed{y/x = r^6}$

Now, let's find the ratio $z/y$ . $z/y = \frac{ar^{15}}{ar^9}$ $z/y = r^{15-9}$

$\boxed{z/y = r^6}$

We see that $y/x = r^6$ and $z/y = r^6$ . Therefore, $y/x = z/y$ . This means $x, y, z$ are in GP.

Answer

(i) The terms $x, y, z$ are in GP.

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Q10

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Q11

The sum of the first three terms of a GP is $\frac{13}{12}$ and their product is $-1$ . Find the common ratio and the terms.

Q12

If the $4^{\text{th}}$ , $10^{\text{th}}$ and $16^{\text{th}}$ terms of a GP are $x$ , $y$ and $z$ respectively, prove that $x, y, z$ are in GP.

Q13

The sum of the first three terms of a geometric progression is 26, and the sum of their squares is 364. Find the terms of the GP.

Q14

Suppose $P_1 = 1$ , $P_2 = 2$ and for $n > 2$ , $P_n = P_1 + P_2 + \dots + P_{n-1} + 1$ . Find the values of $P_1, P_2, \dots, P_8$ . Can you find a simpler recursive formula for $P_n$ ? Can you give an explicit formula?

Q15

Suppose $W_1 = 1$ , $W_2 = 2$ and for $n > 2$ , $W_n = W_1 + W_2 + \dots + W_{n-2} + 2$ . Find the values of $W_1, W_2, \dots, W_8$ . Do you recognise this sequence?