find the g.p. if the sum of its first two terms is 5 and each term is equal to 3 times the sum of its succeeding terms.
answer: `4+1+(1)/(4)+(1)/(6)+...`
solution: n/a
hint: first of all, multiply and divide 9 in both the series after taking common 5 and 0.6 from them respectively. then make a g.p of the form \[10,{{10}^{2}},{{10}^{3}}....\] by writing 9 = 10 – 1, 99 = 100 – 1, and so on. finally, use the formula for the sum of n terms of g.p that is \[\dfrac{a\left( 1-{{r}^{n}} \right)}{\left( 1-r \right)}\].complete step-by-step solution -in this question, we have to find the sum of the n terms of the series.(i) 5 + 55 + 555 + ….(ii) .6 + .66 + .666 + ….let us consider the first seriess = 5 + 55 + 555 + ….. n termsby taking out 5 common from the above series, we get, s = 5 (1 + 11 + 111 + 1111 …. n terms)by multiplying 9 on both the sides of the above equation, we get, 9s = 5 [9 + 99 + 999 …… n times]\[\dfrac{9s}{5}=9+99+999.....\text{ n times}\]we can also write the above equation as, \[\dfrac{9s}{5}=\left( 10-1 \right)+\left( 100-1 \right)+\left( 1000-1 \right)....\text{ n times}\]by simplifying the above equation and taking the term 1 separately, we get, \[\dfrac{9s}{5}=\left( {{10}^{1}}+{{10}^{2}}+{{10}^{3}}.....\text{ n times} \right)-\left( 1+1+1......\text{ n times} \right)\]we know that 1 added n times is equal to n and the nth term of 1, 2, 3…. n. so, we get, \[\dfrac{9s}{5}=\left( {{10}^{1}}+{{10}^{2}}+{{10}^{3}}{{....10}^{n}} \right)-\left( n \right)\]we know that the series of the form \[a,ar,a{{r}^{2}},a{{r}^{3}}.....a{{r}^{n}}\] is in g.p and sum of the n terms of this g.p is given by \[\dfrac{a\left( 1-{{r}^{n}} \right)}{\left( 1-r \right)}\]in the above equation, we can see that \[10,{{10}^{2}},{{10}^{3}}....\] are in g.p with the first term and common ratio as 10. so, we get, \[\dfrac{9s}{5}=\dfrac{10\left( 1-{{\left( 10 \right)}^{n}} \right)}{1-10}-n\]\[\dfrac{9s}{5}=\dfrac{10\left( 1-{{\left( 10 \right)}^{n}} \right)}{-9}-n\]\[\dfrac{9s}{5}=\dfrac{10}{9}\left( {{10}^{n}}-1 \right)-n\]\[\dfrac{9s}{5}=\dfrac{10\left( {{10}^{n}}-1 \right)-9n}{9}\]by multiplying \[\dfrac{5}{9}\] on both the sides of the above equation, we get, \[s=\dfrac{5}{9}\left[ \dfrac{10\left( {{10}^{n}}-1 \right)-9n}{9} \right]\]\[s=\dfrac{50\left( {{10}^{n}}-1 \right)}{81}-\dfrac{5}{9}n\]hence, the sum of the sequence 5 + 55 + 555….. up to n terms is equal to \[\dfrac{50\left( {{10}^{n}}-1 \right)}{81}-\dfrac{5n}{9}\]let us consider the second set.s = 0.6 + 0.66 + 0.666 + ….. n times.by taking out 0.6 common, we can write the sequence as, s = 6 [0.1 + 0.11 + 0.111 ….. n terms]by multiplying and dividing rhs by 9, we get,\[s=\dfrac{6}{9}\left[ 0.9+0.99+0.999.....n\text{ terms} \right]\]\[s=\dfrac{2}{3}\left[ \dfrac{9}{10}+\dfrac{99}{100}+\dfrac{999}{1000}+\dfrac{9999}{10000}.....n\text{ terms} \right]\]we can write the above equation as, \[s=\dfrac{2}{3}\left[ \dfrac{\left( 10-1 \right)}{10}+\dfrac{\left( 100-1 \right)}{100}+\dfrac{\left( 1000-1 \right)}{1000}+\dfrac{\left( 10000-1 \right)}{10000}.....n\text{ terms} \right]\]\[s=\dfrac{2}{3}\left[ \left( 1-\dfrac{1}{10} \right)+\left( 1-\dfrac{1}{100} \right)+\left( 1-\dfrac{1}{1000} \right)+\left( 1-\dfrac{1}{10000} \right).....n\text{ terms} \right]\]\[s=\dfrac{2}{3}\left[ \left( 1+1+1......n\text{ terms} \right)-\left( \dfrac{1}{10}+\dfrac{1}{100}+\dfrac{1}{1000}.+...n\text{ terms} \right) \right]\]here, we can see that \[\dfrac{1}{10}+\dfrac{1}{{{\left( 10 \right)}^{2}}}+\dfrac{1}{{{\left( 10 \right)}^{3}}}+.....n\text{ terms}\] is in gp with \[a=\dfrac{1}{10}\] and \[r=\dfrac{1}{10}\].we know that the sum of n terms of the g.p is \[\dfrac{a\left( 1-{{r}^{n}} \right)}{\left( 1-r \right)}\]. so, we get, \[s=\dfrac{2}{3}\left[ \left( n \right)-\dfrac{\dfrac{1}{10}\left( 1-{{\left( \dfrac{1}{10} \right)}^{n}} \right)}{1-\dfrac{1}{10}} \right]\]\[s=\dfrac{2}{3}\left[ n-\dfrac{\dfrac{1}{10}\left( 1-\dfrac{1}{{{10}^{n}}} \right)}{\dfrac{9}{10}} \right]\]\[s=\dfrac{2}{3}\left[ n-\dfrac{1}{9}\left( 1-\dfrac{1}{{{10}^{n}}} \right) \right]\]we know that \[\dfrac{1}{{{a}^{n}}}={{a}^{-n}}\]. by using this, we get, \[s=\dfrac{2}{3}\left( n-\dfrac{1}{9}\left( 1-{{10}^{-n}} \right) \right)\]so, \[s=\dfrac{2}{3}n-\dfrac{2}{27}\left[ 1-{{\left( 10 \right)}^{-n}} \right]\]hence, we get the sum of the sequence 0.6 + 0.66 + 0.666 + ….. n terms as \[\dfrac{2}{3}n-\dfrac{2}{27}\left[ 1-{{\left( 10 \right)}^{-n}} \right]\].note: in this question, students must note that whenever we are in need to find the sum of series of the type a + aa + aaa + aaaa ….., we always have to make the series of 9 + 99 + 999 + ….. type by multiplying and dividing the whole series by 9 so that we can get a g.p of the type \[10,{{10}^{2}},{{10}^{3}}\]…. also in this question, some students make this mistake of considering 5 + 55 + 555 + …. or 0.6 + 0.66 + 0.666 + …. in g.p which is wrong because we can see that there is no common ratio in these sequences. sum of a g.p. series is represented by the formula, , when |r|<1. ‘sn’ represents the sum of the g.p. series upto nth terms, ‘a’ represents the first term, ‘r’ represents the common ratio and ‘n’ represents the number of terms.here,a = 1r = (ratio between the n term and n-1 term) n terms∴ ⇒∴ a sum of money becomes 3 times in 5 years at simple interest. in how many years will the same sum become 6 times at the same rate of simple interest? use

summation of series of natural numbers -


 

- wherein

sum of first n natural numbers



 

 and

 

summation of series of natural numbers -


 

- wherein

sum of  squares of first n natural numbers



 

 

3+7+14+24+37 15k

let tn=an2+bn+c            because difference is in a.p: [4,7,10,13]

at n=1     (i)

    n=2      (ii)

    n=3        (iii)

    (ii)-(i)        (iv)

    (iii)-(ii)       (v)

    (v)-(iv)      

                    

 

from (i) 

from (ii) 







 



      

 

put n=15   

               

               

               

               

 




option 1)

126

this option is incorrect

option 2)

122

this option is correct

option 3)

81

this option is incorrect

option 4)

119

this option is incorrect write a pair of intergers whose sum gives [readmore]typed answer2no + o2 → 2no2rate=knomo2n2nd order in no → m = 23rd order overall → m + n = 3so 2 + n = 3n = 1rate=k[no]2[o2] let, y = (1 – 2 tan x)(5 + 4 sin x)we have to find dy/dxas we can observe that y is a product of two functions say u and v where,u = (1 – 2tan x) and v = (5 + 4sin x)∴ y = uvas we know that to find the derivative of product of two function we apply product rule of differentiation.by product rule, we have –…equation 1as, u = (1 – 2tan x)∴ ⇒ ⇒ …..equation 2 {∵ }as, v = 5 + 4sin x⇒ ⇒ …equation 3 {∵ }∴ from equation 1, we can find dy/dx∴ using equation 2 & 3, we get –⇒ ⇒ ∵ sin x = tan x cos x , so we get –⇒ hence, 1.5 kg = 1.5 × 1000 g = 1500 g∴ ratio of 1.5 kg to 2500 g= 1500 g : 2500 g = `1500/2500` = `3/5` = 3 : 5 we know that- - - - - (i)taking 3 as common, we get3; solving only bracket term first,here a = 2a, b = 3b; putting values in eq.i}applying }}ans: - 3}note: must memorize cubes upto 12 of three numbers, second is twice the first and is also thrice the third. if the average of the three numbers is 44, the largest number is:

option 1) 24                              option 2) 56option 3) 72               option 4) 108option 5)    36