Progressions - Study Mode
[#1] A square is drawn by joining the mid points of the sides of a given square in the same way and this process continues indefinitely. If a side of the first square is 4 cm, determine the sum of the areas all the square.
Correct Answer
(A) 32 Cm 2
Explanation
Solution: Side of the first square is 4 cm. side of second square = $$2sqrt 2 {kern 1pt} $$ cm. Side of third square = 2 cm. and so on, i.e. 4, 2, $$sqrt 2 $$xa0, $$sqrt 2 $$xa0, 1 ........ Thus, area of these square will be = 16, 8, 4, 2, 1, $$frac{1}{2}$$ .......... Hence, Sum of the area of first, second, third square $$eqalign{
& = 16 + 8 + 4 + 2 + 1 + {kern 1pt} ,...... cr
& = {frac{{16}}{{ {1 - {frac{1}{2}} } }}} cr
& = 32,{kern 1pt} c{m^2} cr} $$
[#2] The first term of an Arithmetic Progression is 22 and the last term is -11. If the sum is 66, the number of terms in the sequence are:
Correct Answer
(B) 12
Explanation
Solution: Number of terms = n (let) First term (a) = 22 Last term (l) = - 11 Sum = 66 Sum of an AP is given by: $$ = { ext{Number}},{ ext{of terms}},, imes $$ xa0xa0 $$ {frac{{ {{ ext{First}},{ ext{term}} + { ext{Last}},{ ext{term}}} }}{2}} $$ $$eqalign{
& 66 = { ext{n}} imes {frac{{ {{ ext{a}} + { ext{l}}} }}{2}} cr
& 66 = { ext{n}} imes frac{{ {22 - 11} }}{2} cr
& 66 = { ext{n}} imes {frac{{11}}{2}} cr
& { ext{n}} = frac{{ {66 imes 2} }}{{11}} cr
& { ext{n}} = 12 cr
& { ext{No}}{ ext{.}},{kern 1pt} { ext{of}},{ ext{terms}} = 12 cr} $$
[#3] Find the n th term of the following sequence :
5 + 55 + 555 + . . . . T n
Correct Answer
(C) $$frac{5}{9} imes left( {{{10}^n} - 1}
ight)$$
Explanation
Solution: We will it through option checking method: $$eqalign{
& {frac{5}{9}} imes left( {{{10}^n} - 1}
ight) cr
& { ext{We}}{kern 1pt} {kern 1pt} { ext{put}}{kern 1pt} {kern 1pt} n = 1, cr
& {frac{5}{9}} imes left( {{{10}^1} - 1}
ight) = 5 cr
& n = 2left( {frac{5}{9}}
ight) imes left( {{{10}^2} - 1}
ight) = 55 cr
& n = 3left( {frac{5}{9}}
ight) imes left( {{{10}^3} - 1}
ight) = 555 cr} $$ It means Option C is satisfying the sequence so the n th term would be $${kern 1pt} {frac{5}{9}} imes left( {{{10}^n} - 1}
ight)$$
[#4] The 2 nd and 8 th term of an arithmetic progression are 17 and -1 respectively. What is the 14 th term?
Correct Answer
(C) -19
Explanation
Solution: $$eqalign{
& {T_2} = a + d = 17,.......,left( 1
ight) cr
& {T_8} = a + 7d = - 1,......,left( 2
ight) cr
& { ext{on solving}}left( 1
ight),{ ext{and}},left( 2
ight) cr
& d = - 3,& ,a = 20 cr
& {T_{14}} = a + 13d cr
& ,,,,,,,,,, = 20 + 13left( { - 3}
ight) cr
& ,,,,,,,,,, = - 19 cr} $$
[#5] The 2 nd and 6 th term of an arithmetic progression are 8 and 20 respectively. What is the 20 th term?
Correct Answer
(D) 62
Explanation
Solution: $$eqalign{
& {T_2} = a + d = 8,.......,left( 1
ight) cr
& {T_6} = a + 5d = 20,......,left( 2
ight) cr
& { ext{on solving}}left( 1
ight),{ ext{and}},left( 2
ight) cr
& d = 3,& ,a = 5 cr
& {T_{20}} = a + 19d cr
& ,,,,,,,,,, = 5 + 19left( 3
ight) cr
& ,,,,,,,,,, = 62 cr} $$