Question

If x^4, x^k, and x^52 are the first second, and eighth terms respectively of a geometric progression, then what is the value of (x²+ 2)?
17
18
20
21
​

Answer 1

If x^4, x^k, and x^52 are the first second, and eighth terms respectively of a geometric progression, then what is the value of (x²+ 2)?

17

18

20

21

Answer:- 18

Answer 2

$\bf {k}^{2} + 2 = 18$

Option (b) is correct.

Given*:

• If ${x}^{ - 4}, \: {x}^{k}, \: and \: {x}^{52}$ are the first second, and eighth terms respectively of a geometric progression.

To find:

• What is the value of (k²+ 2)?
• (a) 17
• (b) 18
• (c) 20
• (d) 21

Solution:

Concept \Formula to be used:

First term of GP is 'a' ,common ratio is 'r'.

General term of GP; $\bf T_n = a {r}^{n - 1}$

Step 1:

Write the given terms.

First term $T_1 = a = {x}^{ - 4}$

and $T_2 = {x}^{k}$

and $T_8 = {x}^{52}$

Step 2:

Write 8th term according to the formula.

$a {r}^{7} = {x}^{52}$

put the value of first term.

${x}^{ - 4} {r}^{7} = {x}^{52}$

or

${r}^{7} = \frac{{x}^{52} }{{x}^{ - 4} }$

or

${r}^{7} = {x}^{52 + 4}$

or

${r}^{7} = {x}^{56}$

or

${r}^{7} = ({x}^{8} )^{7}$

Thus,

$\bf r = {x}^{8}$

Step 3:

Find value of k.

Second term is

$ar = {x}^{k}$

or

put the values of a and r.

${x}^{ - 4} {x}^{8} = {x}^{k}$

or

${x}^{4} = {x}^{k}$

Compare powers,

$\bf k = 4$

Step 4:

Find the value of ${k}^{2} + 2$

Put k=4

So,

${k}^{2} + 2 = ( {4)}^{2} + 2$

or

$= 16 + 2$

or

$= 18$

Thus,

$\bf {k}^{2} + 2 = 18$

Option (b) is correct.

Remark*: Correct question have written in the explanation.

Learn more:

1) The 4th and 7th and the last term of GP are 10 ,80 and 2560 is respectively find the first term and number of terms in G...

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2) If the first term of a G.P is 16 and its sum to infinity is 96/17 then find the common ratio

https://brainly.in/question/12404335

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