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Page 1
Solved Examples on Sequence and
Series
JEE Mains
Q. 1 If ?? , ?? , ?? are distinct positive real in H.P., then the value of the expression,
?? +?? ?? -?? +
?? +?? ?? -?? is equal to
(A) 1
(B) 2
(C) 3
(D) 4
Ans: (B) ?? =
2????
?? +?? [?? , ?? , ?? are in HP]
=
?? + ?? ?? - ?? +
?? + ?? ?? - ?? =
2????
?? + ?? + ?? 2????
?? + ?? - ?? +
2????
?? + ?? + ?? 2????
?? + ?? - ?? =
3???? + ?? 2
???? - ?? 2
+
3???? + ?? 2
???? - ?? 2
=
3?? + ?? ?? - ?? +
( 3?? + ?? )
?? - ?? =
3?? + ?? ?? - ?? -
( 3?? + ?? )
?? - ?? =
2?? - 2?? ?? - ?? = 2
Q. 2 The sum of infinity of the series
?? ?? +
?? ?? +?? +
?? ?? +?? +?? + ? ….. is equal to
(A) 2
(B) ?? /??
(C) 3
(D) None
Ans 2: (A) Given summation is,
1
1
+
1
1+2
+
1
1+2+3
… ?? ?? =
2
?? ( ?? + 1)
? ?? ?? = 2 [
1
?? -
1
?? + 1
]
?? ?? = ?? ?? ?? = 2 [
1
1
-
1
2
+
1
2
-
1
3
+
1
3
…
1
8
] = 2
Q3 If the sum of the first 11 terms of an arithmetical progression equals that of
the first 19 terms, then the sum of its first 30 terms, is
(A) Equal to 0
(B) Equal to -1
(C) Equal to 1
(D) Non unique
Page 2
Solved Examples on Sequence and
Series
JEE Mains
Q. 1 If ?? , ?? , ?? are distinct positive real in H.P., then the value of the expression,
?? +?? ?? -?? +
?? +?? ?? -?? is equal to
(A) 1
(B) 2
(C) 3
(D) 4
Ans: (B) ?? =
2????
?? +?? [?? , ?? , ?? are in HP]
=
?? + ?? ?? - ?? +
?? + ?? ?? - ?? =
2????
?? + ?? + ?? 2????
?? + ?? - ?? +
2????
?? + ?? + ?? 2????
?? + ?? - ?? =
3???? + ?? 2
???? - ?? 2
+
3???? + ?? 2
???? - ?? 2
=
3?? + ?? ?? - ?? +
( 3?? + ?? )
?? - ?? =
3?? + ?? ?? - ?? -
( 3?? + ?? )
?? - ?? =
2?? - 2?? ?? - ?? = 2
Q. 2 The sum of infinity of the series
?? ?? +
?? ?? +?? +
?? ?? +?? +?? + ? ….. is equal to
(A) 2
(B) ?? /??
(C) 3
(D) None
Ans 2: (A) Given summation is,
1
1
+
1
1+2
+
1
1+2+3
… ?? ?? =
2
?? ( ?? + 1)
? ?? ?? = 2 [
1
?? -
1
?? + 1
]
?? ?? = ?? ?? ?? = 2 [
1
1
-
1
2
+
1
2
-
1
3
+
1
3
…
1
8
] = 2
Q3 If the sum of the first 11 terms of an arithmetical progression equals that of
the first 19 terms, then the sum of its first 30 terms, is
(A) Equal to 0
(B) Equal to -1
(C) Equal to 1
(D) Non unique
Ans: ( ?? ) ?? ????
= ?? ????
????
?? [?? ?? + ???? ?? ] =
????
?? [?? ?? + ???? ?? ]
???? ?? = -?????? ?? ?
?? ?? =
-????
?? ?? ????
=
????
?? [?? ?? + ???? ?? ] = ???? [?? +
???? ?? ?? ] = ??
Q4. Let ?? ?? , ?? ?? , ?? ?? …. and ?? ?? , ?? ?? , ?? ?? … are two arithmetic sequence such that ?? ?? = ?? ?? ?
?? ; ?? ?? = ???? and ?
?? =?? ????
? ?? ?? = ?
?? =?? ????
??? ?? Then the value of
?? ?? -?? ?? ?? ?? -?? ?? is
(A) ?? /??
(B) ?? /??
(C) ???? /??
(D) 2
Ans : (C) ?? 2
= 2?? 2
?? 1
+ ?? ?? = 2( ?? 1
+ ?? ?? )
?? ?? = ?? 1
+ 2?? ?? ?
10
2
[2?? 1
+ 9?? ?? ] =
15
2
[2?? 1
+ 14?? ?? ]
? 18?? ?? = 2?? 1
+ 42?? ?? ? 18?? ?? = 2?? ?? - 4?? ?? + 42?? ?? ? 16?? ?? = 38?? ?? ?? ?? ?? ?? =
19
8
Q5. Let ?? ?? '?? ? ?? be the ?? th
term an A.P. with common difference ' ?? ' and all
whose terms are non-zero. If ?? approaches infinity, then the sum
?? ?? ?? ?? ?? +
?? ?? ?? ?? ?? + ? . . +
?? ?? ?? ?? ?? +?? will approach
(A)
?? ?? ?? ??
(B)
?? ?? ?? ??
(C)
?? ?? ?? ?? ??
(D) ?? ?? ??
Ans: (A) The given expression is equal to
1
?? (
1
?? 1
-
1
?? 2
+
1
?? 2
-
1
?? 3
+ ? ) =
1
?? 1
??
Page 3
Solved Examples on Sequence and
Series
JEE Mains
Q. 1 If ?? , ?? , ?? are distinct positive real in H.P., then the value of the expression,
?? +?? ?? -?? +
?? +?? ?? -?? is equal to
(A) 1
(B) 2
(C) 3
(D) 4
Ans: (B) ?? =
2????
?? +?? [?? , ?? , ?? are in HP]
=
?? + ?? ?? - ?? +
?? + ?? ?? - ?? =
2????
?? + ?? + ?? 2????
?? + ?? - ?? +
2????
?? + ?? + ?? 2????
?? + ?? - ?? =
3???? + ?? 2
???? - ?? 2
+
3???? + ?? 2
???? - ?? 2
=
3?? + ?? ?? - ?? +
( 3?? + ?? )
?? - ?? =
3?? + ?? ?? - ?? -
( 3?? + ?? )
?? - ?? =
2?? - 2?? ?? - ?? = 2
Q. 2 The sum of infinity of the series
?? ?? +
?? ?? +?? +
?? ?? +?? +?? + ? ….. is equal to
(A) 2
(B) ?? /??
(C) 3
(D) None
Ans 2: (A) Given summation is,
1
1
+
1
1+2
+
1
1+2+3
… ?? ?? =
2
?? ( ?? + 1)
? ?? ?? = 2 [
1
?? -
1
?? + 1
]
?? ?? = ?? ?? ?? = 2 [
1
1
-
1
2
+
1
2
-
1
3
+
1
3
…
1
8
] = 2
Q3 If the sum of the first 11 terms of an arithmetical progression equals that of
the first 19 terms, then the sum of its first 30 terms, is
(A) Equal to 0
(B) Equal to -1
(C) Equal to 1
(D) Non unique
Ans: ( ?? ) ?? ????
= ?? ????
????
?? [?? ?? + ???? ?? ] =
????
?? [?? ?? + ???? ?? ]
???? ?? = -?????? ?? ?
?? ?? =
-????
?? ?? ????
=
????
?? [?? ?? + ???? ?? ] = ???? [?? +
???? ?? ?? ] = ??
Q4. Let ?? ?? , ?? ?? , ?? ?? …. and ?? ?? , ?? ?? , ?? ?? … are two arithmetic sequence such that ?? ?? = ?? ?? ?
?? ; ?? ?? = ???? and ?
?? =?? ????
? ?? ?? = ?
?? =?? ????
??? ?? Then the value of
?? ?? -?? ?? ?? ?? -?? ?? is
(A) ?? /??
(B) ?? /??
(C) ???? /??
(D) 2
Ans : (C) ?? 2
= 2?? 2
?? 1
+ ?? ?? = 2( ?? 1
+ ?? ?? )
?? ?? = ?? 1
+ 2?? ?? ?
10
2
[2?? 1
+ 9?? ?? ] =
15
2
[2?? 1
+ 14?? ?? ]
? 18?? ?? = 2?? 1
+ 42?? ?? ? 18?? ?? = 2?? ?? - 4?? ?? + 42?? ?? ? 16?? ?? = 38?? ?? ?? ?? ?? ?? =
19
8
Q5. Let ?? ?? '?? ? ?? be the ?? th
term an A.P. with common difference ' ?? ' and all
whose terms are non-zero. If ?? approaches infinity, then the sum
?? ?? ?? ?? ?? +
?? ?? ?? ?? ?? + ? . . +
?? ?? ?? ?? ?? +?? will approach
(A)
?? ?? ?? ??
(B)
?? ?? ?? ??
(C)
?? ?? ?? ?? ??
(D) ?? ?? ??
Ans: (A) The given expression is equal to
1
?? (
1
?? 1
-
1
?? 2
+
1
?? 2
-
1
?? 3
+ ? ) =
1
?? 1
??
Q6. The sum ?
?? =?? 8
?(
?? ?? ?? +?? ) is equal to
(A) ?? /??
(B) ?? /??
(C) ?? /??
(D) ½
Ans: (C) ?
?? =1
8
?
?? ?? 4
+ 4
=
1
?? (?? 2
+
4
?? 2
+ 4 - 4)
= ?
1
?? [(?? +
2
?? )
2
- 4]
= ?
1
?? [?? +
2
?? - 2] [?? +
2
?? + 2]
= ?
?? ( ?? 2
+ 2 - 2?? ) ( ?? 2
+ 2?? + 2)
=
1
4
? [
1
?? 2
- 2?? + 2
-
1
?? 2
+ 2?? + 2
]
=
1
4
?
?? =1
8
?[
1
( ?? - 1)
2
+ 1
-
1
( ?? + 1)
2
+ 1
]
=
1
4
(1 +
1
2
+ ?
?? =1
8
?[
1
( ?? + 1)
2
+ 1
-
1
( ?? + 1)
2
+ 1
]) =
3
8
Q7. If ?? ? ?? and ( ???? ?? ?? )+ ( ???? ?? ?? )
?? + ( ???? ?? ?? )
?? + ? . . = ?? [???? ?? +( ???? ?? )
?? + ( ???? ?? )
?? +
( ???? ?? )
?? + ? … . ], then ' ?? ' is equal to
(A) ?? ?? /??
(B) ?? ?? /??
(C) ?? ?? ?? /??
(D) ?? ?? /??
Ans: (D) ln ?? 2
+ ( ln ?? 2
)
2
+ ( ln ?? 2
)
3
… . .
= 3{ln ?? + ( ln ?? )
2
+ ( ln ?? )
3
+ ? . }
?
2ln ?? 1 - 2ln ?? =
3ln ?? 1 - ln ?? ? 2 - 2ln ?? = 3 - 6ln ?? ? 1 = 4ln ?? ? ?? = ?? 1/4
Page 4
Solved Examples on Sequence and
Series
JEE Mains
Q. 1 If ?? , ?? , ?? are distinct positive real in H.P., then the value of the expression,
?? +?? ?? -?? +
?? +?? ?? -?? is equal to
(A) 1
(B) 2
(C) 3
(D) 4
Ans: (B) ?? =
2????
?? +?? [?? , ?? , ?? are in HP]
=
?? + ?? ?? - ?? +
?? + ?? ?? - ?? =
2????
?? + ?? + ?? 2????
?? + ?? - ?? +
2????
?? + ?? + ?? 2????
?? + ?? - ?? =
3???? + ?? 2
???? - ?? 2
+
3???? + ?? 2
???? - ?? 2
=
3?? + ?? ?? - ?? +
( 3?? + ?? )
?? - ?? =
3?? + ?? ?? - ?? -
( 3?? + ?? )
?? - ?? =
2?? - 2?? ?? - ?? = 2
Q. 2 The sum of infinity of the series
?? ?? +
?? ?? +?? +
?? ?? +?? +?? + ? ….. is equal to
(A) 2
(B) ?? /??
(C) 3
(D) None
Ans 2: (A) Given summation is,
1
1
+
1
1+2
+
1
1+2+3
… ?? ?? =
2
?? ( ?? + 1)
? ?? ?? = 2 [
1
?? -
1
?? + 1
]
?? ?? = ?? ?? ?? = 2 [
1
1
-
1
2
+
1
2
-
1
3
+
1
3
…
1
8
] = 2
Q3 If the sum of the first 11 terms of an arithmetical progression equals that of
the first 19 terms, then the sum of its first 30 terms, is
(A) Equal to 0
(B) Equal to -1
(C) Equal to 1
(D) Non unique
Ans: ( ?? ) ?? ????
= ?? ????
????
?? [?? ?? + ???? ?? ] =
????
?? [?? ?? + ???? ?? ]
???? ?? = -?????? ?? ?
?? ?? =
-????
?? ?? ????
=
????
?? [?? ?? + ???? ?? ] = ???? [?? +
???? ?? ?? ] = ??
Q4. Let ?? ?? , ?? ?? , ?? ?? …. and ?? ?? , ?? ?? , ?? ?? … are two arithmetic sequence such that ?? ?? = ?? ?? ?
?? ; ?? ?? = ???? and ?
?? =?? ????
? ?? ?? = ?
?? =?? ????
??? ?? Then the value of
?? ?? -?? ?? ?? ?? -?? ?? is
(A) ?? /??
(B) ?? /??
(C) ???? /??
(D) 2
Ans : (C) ?? 2
= 2?? 2
?? 1
+ ?? ?? = 2( ?? 1
+ ?? ?? )
?? ?? = ?? 1
+ 2?? ?? ?
10
2
[2?? 1
+ 9?? ?? ] =
15
2
[2?? 1
+ 14?? ?? ]
? 18?? ?? = 2?? 1
+ 42?? ?? ? 18?? ?? = 2?? ?? - 4?? ?? + 42?? ?? ? 16?? ?? = 38?? ?? ?? ?? ?? ?? =
19
8
Q5. Let ?? ?? '?? ? ?? be the ?? th
term an A.P. with common difference ' ?? ' and all
whose terms are non-zero. If ?? approaches infinity, then the sum
?? ?? ?? ?? ?? +
?? ?? ?? ?? ?? + ? . . +
?? ?? ?? ?? ?? +?? will approach
(A)
?? ?? ?? ??
(B)
?? ?? ?? ??
(C)
?? ?? ?? ?? ??
(D) ?? ?? ??
Ans: (A) The given expression is equal to
1
?? (
1
?? 1
-
1
?? 2
+
1
?? 2
-
1
?? 3
+ ? ) =
1
?? 1
??
Q6. The sum ?
?? =?? 8
?(
?? ?? ?? +?? ) is equal to
(A) ?? /??
(B) ?? /??
(C) ?? /??
(D) ½
Ans: (C) ?
?? =1
8
?
?? ?? 4
+ 4
=
1
?? (?? 2
+
4
?? 2
+ 4 - 4)
= ?
1
?? [(?? +
2
?? )
2
- 4]
= ?
1
?? [?? +
2
?? - 2] [?? +
2
?? + 2]
= ?
?? ( ?? 2
+ 2 - 2?? ) ( ?? 2
+ 2?? + 2)
=
1
4
? [
1
?? 2
- 2?? + 2
-
1
?? 2
+ 2?? + 2
]
=
1
4
?
?? =1
8
?[
1
( ?? - 1)
2
+ 1
-
1
( ?? + 1)
2
+ 1
]
=
1
4
(1 +
1
2
+ ?
?? =1
8
?[
1
( ?? + 1)
2
+ 1
-
1
( ?? + 1)
2
+ 1
]) =
3
8
Q7. If ?? ? ?? and ( ???? ?? ?? )+ ( ???? ?? ?? )
?? + ( ???? ?? ?? )
?? + ? . . = ?? [???? ?? +( ???? ?? )
?? + ( ???? ?? )
?? +
( ???? ?? )
?? + ? … . ], then ' ?? ' is equal to
(A) ?? ?? /??
(B) ?? ?? /??
(C) ?? ?? ?? /??
(D) ?? ?? /??
Ans: (D) ln ?? 2
+ ( ln ?? 2
)
2
+ ( ln ?? 2
)
3
… . .
= 3{ln ?? + ( ln ?? )
2
+ ( ln ?? )
3
+ ? . }
?
2ln ?? 1 - 2ln ?? =
3ln ?? 1 - ln ?? ? 2 - 2ln ?? = 3 - 6ln ?? ? 1 = 4ln ?? ? ?? = ?? 1/4
Q8. Let ?? ?? , ?? ?? , ?? ?? ….. and ?? ?? , ?? ?? , ?? ?? …. are two arithmetic sequence such that ?? ?? = ?? ?? ?
?? ; ?? ?? = ?? ?? ?? and ?
?? =?? ????
??? ?? = ?
?? =?? ????
??? ?? . Then the value of
?? ?? -?? ?? ?? ?? -?? ?? is
(A) ?? /??
(B) ?? /??
(C) ???? /??
(D) 2
Ans: (C) ?? 2
= 2?? 2
?? 1
+ ?? ?? = 2( ?? 1
+ ?? ?? )
?? ?? = ?? 1
+ 2?? ?? ?
10
2
[2?? 1
+ 9?? ?? ] =
15
2
[2?? 1
+ 14?? ?? ]
? 18?? ?? = 2?? 1
+ 42?? ?? ? 18?? ?? = 2?? ?? - 4?? ?? + 42?? ?? ? 16?? ?? = 38?? ?? ?? ?? ?? ?? =
19
8
Q9 The sum of the first three terms of an increasing G.P. is 21 and the sum of
their squares is 189 . Then the sum of its first ?? term is
(A) ?? ( ?? ?? - ?? )
(B) ???? (?? -
?? ?? ?? )
(C) ?? (?? -
?? ?? ?? )
(D) ?? ( ?? ?? - ?? )
Ans: (A) ?? + ???? + ?? ?? 2
= 21
?? 2
+ ?? 2
?? 2
+ ?? 2
?? 4
= 189
Squaring equation (i) & then dividing by (ii)
?
?? 2
( 1 + ?? + ?? 2
)
2
?? 2
( 1 + ?? 2
+ ?? 4
)
=
441
189
?
( 1 + ?? + ?? 2
) ( 1 + ?? + ?? 2
)
( 1 + ?? + ?? 2
) ( 1 - ?? + ?? 2
)
=
441
189
? 2?? 2
- 5?? + 2 = 0
? ?? = 2,
1
2
? ?? = 3,12
GP is 3,6,12… ?? ?? = 3
( 2
?? -1)
2-1
= 3( 2
?? - 1) Hence, (A) is the correct choice
Page 5
Solved Examples on Sequence and
Series
JEE Mains
Q. 1 If ?? , ?? , ?? are distinct positive real in H.P., then the value of the expression,
?? +?? ?? -?? +
?? +?? ?? -?? is equal to
(A) 1
(B) 2
(C) 3
(D) 4
Ans: (B) ?? =
2????
?? +?? [?? , ?? , ?? are in HP]
=
?? + ?? ?? - ?? +
?? + ?? ?? - ?? =
2????
?? + ?? + ?? 2????
?? + ?? - ?? +
2????
?? + ?? + ?? 2????
?? + ?? - ?? =
3???? + ?? 2
???? - ?? 2
+
3???? + ?? 2
???? - ?? 2
=
3?? + ?? ?? - ?? +
( 3?? + ?? )
?? - ?? =
3?? + ?? ?? - ?? -
( 3?? + ?? )
?? - ?? =
2?? - 2?? ?? - ?? = 2
Q. 2 The sum of infinity of the series
?? ?? +
?? ?? +?? +
?? ?? +?? +?? + ? ….. is equal to
(A) 2
(B) ?? /??
(C) 3
(D) None
Ans 2: (A) Given summation is,
1
1
+
1
1+2
+
1
1+2+3
… ?? ?? =
2
?? ( ?? + 1)
? ?? ?? = 2 [
1
?? -
1
?? + 1
]
?? ?? = ?? ?? ?? = 2 [
1
1
-
1
2
+
1
2
-
1
3
+
1
3
…
1
8
] = 2
Q3 If the sum of the first 11 terms of an arithmetical progression equals that of
the first 19 terms, then the sum of its first 30 terms, is
(A) Equal to 0
(B) Equal to -1
(C) Equal to 1
(D) Non unique
Ans: ( ?? ) ?? ????
= ?? ????
????
?? [?? ?? + ???? ?? ] =
????
?? [?? ?? + ???? ?? ]
???? ?? = -?????? ?? ?
?? ?? =
-????
?? ?? ????
=
????
?? [?? ?? + ???? ?? ] = ???? [?? +
???? ?? ?? ] = ??
Q4. Let ?? ?? , ?? ?? , ?? ?? …. and ?? ?? , ?? ?? , ?? ?? … are two arithmetic sequence such that ?? ?? = ?? ?? ?
?? ; ?? ?? = ???? and ?
?? =?? ????
? ?? ?? = ?
?? =?? ????
??? ?? Then the value of
?? ?? -?? ?? ?? ?? -?? ?? is
(A) ?? /??
(B) ?? /??
(C) ???? /??
(D) 2
Ans : (C) ?? 2
= 2?? 2
?? 1
+ ?? ?? = 2( ?? 1
+ ?? ?? )
?? ?? = ?? 1
+ 2?? ?? ?
10
2
[2?? 1
+ 9?? ?? ] =
15
2
[2?? 1
+ 14?? ?? ]
? 18?? ?? = 2?? 1
+ 42?? ?? ? 18?? ?? = 2?? ?? - 4?? ?? + 42?? ?? ? 16?? ?? = 38?? ?? ?? ?? ?? ?? =
19
8
Q5. Let ?? ?? '?? ? ?? be the ?? th
term an A.P. with common difference ' ?? ' and all
whose terms are non-zero. If ?? approaches infinity, then the sum
?? ?? ?? ?? ?? +
?? ?? ?? ?? ?? + ? . . +
?? ?? ?? ?? ?? +?? will approach
(A)
?? ?? ?? ??
(B)
?? ?? ?? ??
(C)
?? ?? ?? ?? ??
(D) ?? ?? ??
Ans: (A) The given expression is equal to
1
?? (
1
?? 1
-
1
?? 2
+
1
?? 2
-
1
?? 3
+ ? ) =
1
?? 1
??
Q6. The sum ?
?? =?? 8
?(
?? ?? ?? +?? ) is equal to
(A) ?? /??
(B) ?? /??
(C) ?? /??
(D) ½
Ans: (C) ?
?? =1
8
?
?? ?? 4
+ 4
=
1
?? (?? 2
+
4
?? 2
+ 4 - 4)
= ?
1
?? [(?? +
2
?? )
2
- 4]
= ?
1
?? [?? +
2
?? - 2] [?? +
2
?? + 2]
= ?
?? ( ?? 2
+ 2 - 2?? ) ( ?? 2
+ 2?? + 2)
=
1
4
? [
1
?? 2
- 2?? + 2
-
1
?? 2
+ 2?? + 2
]
=
1
4
?
?? =1
8
?[
1
( ?? - 1)
2
+ 1
-
1
( ?? + 1)
2
+ 1
]
=
1
4
(1 +
1
2
+ ?
?? =1
8
?[
1
( ?? + 1)
2
+ 1
-
1
( ?? + 1)
2
+ 1
]) =
3
8
Q7. If ?? ? ?? and ( ???? ?? ?? )+ ( ???? ?? ?? )
?? + ( ???? ?? ?? )
?? + ? . . = ?? [???? ?? +( ???? ?? )
?? + ( ???? ?? )
?? +
( ???? ?? )
?? + ? … . ], then ' ?? ' is equal to
(A) ?? ?? /??
(B) ?? ?? /??
(C) ?? ?? ?? /??
(D) ?? ?? /??
Ans: (D) ln ?? 2
+ ( ln ?? 2
)
2
+ ( ln ?? 2
)
3
… . .
= 3{ln ?? + ( ln ?? )
2
+ ( ln ?? )
3
+ ? . }
?
2ln ?? 1 - 2ln ?? =
3ln ?? 1 - ln ?? ? 2 - 2ln ?? = 3 - 6ln ?? ? 1 = 4ln ?? ? ?? = ?? 1/4
Q8. Let ?? ?? , ?? ?? , ?? ?? ….. and ?? ?? , ?? ?? , ?? ?? …. are two arithmetic sequence such that ?? ?? = ?? ?? ?
?? ; ?? ?? = ?? ?? ?? and ?
?? =?? ????
??? ?? = ?
?? =?? ????
??? ?? . Then the value of
?? ?? -?? ?? ?? ?? -?? ?? is
(A) ?? /??
(B) ?? /??
(C) ???? /??
(D) 2
Ans: (C) ?? 2
= 2?? 2
?? 1
+ ?? ?? = 2( ?? 1
+ ?? ?? )
?? ?? = ?? 1
+ 2?? ?? ?
10
2
[2?? 1
+ 9?? ?? ] =
15
2
[2?? 1
+ 14?? ?? ]
? 18?? ?? = 2?? 1
+ 42?? ?? ? 18?? ?? = 2?? ?? - 4?? ?? + 42?? ?? ? 16?? ?? = 38?? ?? ?? ?? ?? ?? =
19
8
Q9 The sum of the first three terms of an increasing G.P. is 21 and the sum of
their squares is 189 . Then the sum of its first ?? term is
(A) ?? ( ?? ?? - ?? )
(B) ???? (?? -
?? ?? ?? )
(C) ?? (?? -
?? ?? ?? )
(D) ?? ( ?? ?? - ?? )
Ans: (A) ?? + ???? + ?? ?? 2
= 21
?? 2
+ ?? 2
?? 2
+ ?? 2
?? 4
= 189
Squaring equation (i) & then dividing by (ii)
?
?? 2
( 1 + ?? + ?? 2
)
2
?? 2
( 1 + ?? 2
+ ?? 4
)
=
441
189
?
( 1 + ?? + ?? 2
) ( 1 + ?? + ?? 2
)
( 1 + ?? + ?? 2
) ( 1 - ?? + ?? 2
)
=
441
189
? 2?? 2
- 5?? + 2 = 0
? ?? = 2,
1
2
? ?? = 3,12
GP is 3,6,12… ?? ?? = 3
( 2
?? -1)
2-1
= 3( 2
?? - 1) Hence, (A) is the correct choice
Q. 10 If ?? ? ?? and ( ???? ?? ?? )+ ( ???? ?? ?? )
?? + ( ???? ?? ?? )
?? + ? . . = ?? [???? ?? +( ???? ?? )
?? + ( ???? ?? )
?? +
( ???? ?? )
?? + ? . . ], then ' ?? ' is equal to
(A) ?? ?? /??
(B) ?? ?? /??
(C) ?? ?? ?? /??
(D) ?? ?? /??
Ans: (B) ?? ?? =
?? 2
( ?? +1)
2
4
?? 2
?? ?? =
1
4
( ?? + 1)
2
?? ?? =
1
4
[?? 2
+ 2?? + 1]
?? ?? = ? ?
?? ?? =1
??? ?? ?? ?? =
1
4
[
?? ( ?? + 1) ( 2?? + 1)
6
+ ?? ( ?? + 1)+ ?? ]
?? = 9
?? 9
=
1
4
[
9 × 10 × 19
6
+ 9 × 10 + 9] =
1
4
[285 + 90 + 9] =
384
4
= 96
Q11. If ?
?? =?? ?? ??????? (
?? ?? ?? ?? -?? ) = ?? ?????? (
?? ?? +?? ?? ?? -?? ), then ?? is equal to
(A) ?? + ??
(B) ??
(C) ?? /??
(D)
?? +?? ??
Ans: (C)
We have
? ?
?? ?? =1
?log (
?? ?? ?? ?? -1
) = ? ?
?? ?? =1
?( log ?? ?? - log ?? ?? -1
)
= ? ?
?? ?? =1
?( ?? log ?? - ( ?? - 1) log ?? ) =
?? ( ?? + 1)
2
log ?? -
( ?? - 1) ?? 2
log ?? =
?? 2
[( ?? + 1) log ?? - ( ?? - 1) log ?? ]
=
?? 2
[log
?? ?? +1
?? ?? -1
]
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