Question
If l, m, n are the
terms of a G.P which are +ve, then ![open vertical bar table attributes columnalign left end attributes row cell l o g invisible function application l end cell p 1 row cell l o g invisible function application m end cell q 1 row cell l o g invisible function application n end cell r 1 end table close vertical bar equals](data:image/png;base64,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)
- 3
- 2
- 1
- 0
Hint:
The formula for the nth term of the geometric progression is:
an = arn-1
where, a is the first term
- r is the common ratio
- n is the number of the term which we want to find.
The correct answer is: 0
Given : l, m, n are the
terms of a G.P
We know that in GP
![a subscript n space equals space a r to the power of n minus 1 end exponent](data:image/png;base64,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)
where,
- a is the first term
- r is the common ratio
- n is the number of the term which we want to find.
Using this :
![p to the power of t h end exponent space t e r m space equals space l space equals space space equals space a r to the power of P minus 1 end exponent
q to the power of t h end exponent space t e r m space equals space m space equals space space equals space a r to the power of Q minus 1 end exponent
r to the power of t h end exponent space t e r m space equals space n space equals space space equals space a r to the power of R minus 1 end exponent](data:image/png;base64,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)
Substituting these values in :
![rightwards double arrow open vertical bar table attributes columnalign left left left end attributes row cell log invisible function application l end cell p 1 row cell log invisible function application m end cell q 1 row cell log invisible function application n end cell r 1 end table close vertical bar equals open vertical bar table attributes columnalign left left left end attributes row cell log left parenthesis a r to the power of P minus 1 end exponent right parenthesis end cell p 1 row cell log left parenthesis a r to the power of Q minus 1 end exponent right parenthesis end cell q 1 row cell log left parenthesis a r to the power of R minus 1 end exponent right parenthesis end cell r 1 end table close vertical bar](data:image/png;base64,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)
We know that
![rightwards double arrow log open parentheses a b close parentheses space equals space log a space plus log b space a n d
rightwards double arrow log a to the power of m space equals space m log a](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAOIAAAArCAYAAABl91SrAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAU2v5G4wAABTlJREFUeNrtnG+EVkscx4+VdSWRtR5JIkmSLFeSJNGL5MqKJL1KrOtKrsRKklyxetGL1ZskSXJZK8lKZK0kiSS5ci1Xcl1ZsZIkK7b56XvcxzTnOTPzzJkzz9nvh581s+fffGd+8+8555dlvc9vysYquO4Yrk0IKWGrsqcVXv8J7kFIz7IY4R7iKEOe535W1ldyzM/KHjdYv6a3D9t6rpPKn69qoYfgiD5sUPbK8tjHcEg6Yu+Vz6We6yDK81Ut9CVlJzzPHVb2p+WxJytagzbRERcTK59LPdeB9/OdVrayC6F/UfZa2QL+Hig4d4ey58q+KPtH2THD9R4o22U4V4b5C8rmcJ+XytZpx5xXdhbHvcN9ZpStLniW6UQadUj9bLWK6Yihy+dSz77tSHyipewGppnz6Lx1VigbV/YBzzGp7IqyUR+hpVBvle30EHo7RNuC9Bakd2jHbUJ+7mTrld02XO+jsn7DfadQ4FUQcwI9TzsTuMdo23GXC3qnftzLtnGVmW9DDa2frVaxHLGK8rnUs287Ogkn3I3j1sDRlmtOPYMOow+D2Xk883A3jUnsYskiUxdmEj2e3gPe1fKkUCMW1/tqOOYIxGnnprL9Wt6sss1a3soODreQwIgYWj9brWI5Yujy+dSzTzsyPYuMigNt6XMFI98sHNeb6yj4bQehTSOYabR5j17IxxFnDD3oI21K0YcphM5PiTtiaP1stKpyBlB1+Xzq2bUdfTKcu8xw338xNbV5PucRccxxRFy0bOSLXVScvhVsEms7hLZdC6YyNQ2tn41WMUfE0OVzreeQ7ag9fwhOnFmeX/ka0bbHK/pdRb/eQ8NzvNfS8v8XhmnHDcP1T2EKkepmTWj9bLSK6Yihy+daz6HakZ5/UNkti+Ocd01XdLEG0HfBZKF6x7Abqk8LVhmuZ/r54j9MPdrn5pPaMeNYNOsV/qpgN+0E7pXCGjGkfjZaxV4jhiyfaz37tKPjhnP1/GHDejODc47U0ZC2YQcrXzxvRVof1WRBLD+it9Dz7VP2DDtRmTbk6z/o/6HsKs5bhzXsfe2YO1gk70V6DfKOFZQjlR/0Q+tno1VMRwxdPtd69mlHpo0tPV92T19n/78BJm1pCs6+v46GlO+C/Y15/19Z8dbtcTzoAkTeVrBOe5r9+C7oWfRA59CrzWn3mUPPO4sNn5cdniO1V9xC61emVS+Xz6WeMw9t5rVRs1N+Cw76BW12T4fzk6ZVsE6zeel7sIv7NuWl75blOnew4eXrhsFsCXIPvVyG+fy9DlOKX7NqXkG7FGsOX7N+LB8p5BAW1fkrTL9TEurH9kEIIYQQQgghhBBCCCGEEEIIIYQQQghpJvJG/9Hs+8ed8sa+hNLPXxSWj0D5niJpLKnE5ZRvyuQzGAm/JxHA5GsKiTIgwYA2Ij1PfQkdsVryMH0DHdIz1JfQEatFjw1Slqa+JHlSivSdTzvLojFLDJERh3QsJMDsKNaq4yiDRKPOwydIHJbLmDbL1+dnEtWX1EBKkb4Fm2jMegyRsnRZmUKFS5yAMz7HqLwMjfxN9v1jVwnBdxj5a+E0rQT1JRVOeXoh0rdtNGY9NkhZOhYSS2U6+zFYrsRiuVaQvzpBfUmNpBDpO0Sk6rrIozwvd8ynvhwRk4v0XUWk6lhT06KdWpf8JupLIqwReyFSdSxso0R3yqe+S5iUIn1XEak6FkU7tS751Jd4OWIvRKqOhW2U6E751Jd4OaLQpEjV3eASJboon/qS6DQ9UjX1JUnCSM7UlyQAIzlT3yXFN9oe1WKcFR+NAAABzXRFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtbz4mI3gyMUQyOzwvbW8+PG1pPmxvZzwvbWk+PG1mZW5jZWQ+PG1yb3c+PG1pPmE8L21pPjxtaT5iPC9taT48L21yb3c+PC9tZmVuY2VkPjxtbz4mI3hBMDs8L21vPjxtbz49PC9tbz48bW8+JiN4QTA7PC9tbz48bWk+bG9nPC9taT48bWk+YTwvbWk+PG1vPiYjeEEwOzwvbW8+PG1vPis8L21vPjxtaT5sb2c8L21pPjxtaT5iPC9taT48bW8+JiN4QTA7PC9tbz48bWk+YTwvbWk+PG1pPm48L21pPjxtaT5kPC9taT48bXNwYWNlIGxpbmVicmVhaz0ibmV3bGluZSIvPjxtbz4mI3gyMUQyOzwvbW8+PG1pPmxvZzwvbWk+PG1zdXA+PG1pPmE8L21pPjxtaT5tPC9taT48L21zdXA+PG1vPiYjeEEwOzwvbW8+PG1vPj08L21vPjxtbz4mI3hBMDs8L21vPjxtaT5tPC9taT48bWk+bG9nPC9taT48bWk+YTwvbWk+PC9tYXRoPojgueAAAAAASUVORK5CYII=)
Using these formulas and further simplifying
![rightwards double arrow open vertical bar table attributes columnalign left left left end attributes row cell log a space plus left parenthesis P minus 1 right parenthesis log space r right parenthesis end cell p 1 row cell log a space plus left parenthesis Q minus 1 right parenthesis log space r right parenthesis end cell q 1 row cell log a space plus space left parenthesis R minus 1 right parenthesis log space r right parenthesis end cell r 1 end table close vertical bar
R subscript 2 space rightwards arrow R subscript 2 space minus space R subscript 1 space a n d space R subscript 3 rightwards arrow space R subscript 3 space minus space R subscript 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)
![rightwards double arrow open vertical bar table attributes columnalign left left left end attributes row cell log a space plus left parenthesis P minus 1 right parenthesis log space r right parenthesis end cell p 1 row cell left parenthesis Q minus P right parenthesis log space r end cell cell q space minus p end cell 0 row cell left parenthesis R minus P right parenthesis log space r end cell cell r space minus space p end cell 0 end table close vertical bar
T a k i n g space o u t space c o m m o n space f r o m space R subscript 2 space a n d space R subscript 3
rightwards double arrow left parenthesis Q minus P right parenthesis left parenthesis R minus P right parenthesis open vertical bar table attributes columnalign left left left end attributes row cell log a space plus left parenthesis P minus 1 right parenthesis log space r right parenthesis end cell p 1 row cell log space r end cell 1 0 row cell log space r end cell 1 0 end table close vertical bar
I f space 2 space r o w s space a r e space e q u a l space t h e n space t h e i r space d e t e r m i n a n t space i s space 0
rightwards double arrow left parenthesis Q minus P right parenthesis left parenthesis R minus P right parenthesis open vertical bar table attributes columnalign left left left end attributes row cell log a space plus left parenthesis P minus 1 right parenthesis log space r right parenthesis end cell p 1 row cell log space r end cell 1 0 row cell log space r end cell 1 0 end table close vertical bar space equals space 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)
Thus,
0
Related Questions to study
The relationship between force and position is shown in the figure given (in one dimensional case) calculate the work done by the force in displacing a body from x=0 cm to x=5 cm
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)
The relationship between force and position is shown in the figure given (in one dimensional case) calculate the work done by the force in displacing a body from x=0 cm to x=5 cm
![](data:image/png;base64,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)
Fifth term of G.P is 2 The product of its first nine terms is
Here note that the fifth term is having fourth power of 2 and not fifth power. We need not to find all nine terms separately; only finding the product is enough because that product will then be written in the form of the term that is known. Terms in a G.P. are having a common ratio in between. That’s why the power of r is increasing as the terms are increasing.
Fifth term of G.P is 2 The product of its first nine terms is
Here note that the fifth term is having fourth power of 2 and not fifth power. We need not to find all nine terms separately; only finding the product is enough because that product will then be written in the form of the term that is known. Terms in a G.P. are having a common ratio in between. That’s why the power of r is increasing as the terms are increasing.