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First Year Math Sequences and Series Q.53 A number H is said to be the harmonic mean between two numbers a and b if a H b form(a) H.P.(b) A.P.(c) G.P.(d) harmonic


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Q.53 A number H is said to be the harmonic mean between two numbers a and b if a H b form(a) H.P.(b) A.P.(c) G.P.(d) harmonic

2. Sum the seriesii) \frac{3}{\sqrt{2}}+2 \sqrt{2}+\frac{5}{\sqrt{2}}+\ldots .+a_{13}
2. Sum the seriesii)  \frac{3}{\sqrt{2}}+2 \sqrt{2}+\frac{5}{\sqrt{2}}+\ldots .+a_{13}

2. Sum the seriesii) \frac{3}{\sqrt{2}}+2 \sqrt{2}+\frac{5}{\sqrt{2}}+\ldots .+a_{13}

18. The sum of the first n terms of two A.P.s are as 13-7 n: 3 n+1 . Find the ratio of their first terms and also of their second terms.
18. The sum of the first  n  terms of two A.P.s are as  13-7 n: 3 n+1 . Find the ratio of their first terms and also of their second terms.

18. The sum of the first n terms of two A.P.s are as 13-7 n: 3 n+1 . Find the ratio of their first terms and also of their second terms.

3. A man borrows Rs. 1100 and agree to repay with a total interest of Rs. 230 in 14 installments each installment being less than the preceding by Rs. 10 . What should be his first installment?
3. A man borrows Rs. 1100 and agree to repay with a total interest of Rs. 230 in 14 installments each installment being less than the preceding by Rs. 10 . What should be his first installment?

3. A man borrows Rs. 1100 and agree to repay with a total interest of Rs. 230 in 14 installments each installment being less than the preceding by Rs. 10 . What should be his first installment?

11. What distance will a ball travel before coming to rest if it is dropped from a height of 75 meters and after each fall it rebounds \frac{2}{5} of the distance it fell?
11. What distance will a ball travel before coming to rest if it is dropped from a height of 75 meters and after each fall it rebounds  \frac{2}{5}  of the distance it fell?

11. What distance will a ball travel before coming to rest if it is dropped from a height of 75 meters and after each fall it rebounds \frac{2}{5} of the distance it fell?

8. If S_{2} S_{3} S_{5} are the sums of 2 n 3 n 5 n terms of an A.P. show that S_{5}=5\left(S_{3}-S_{2}\right)
8. If  S_{2} S_{3} S_{5}  are the sums of  2 n 3 n 5 n  terms of an A.P. show that  S_{5}=5\left(S_{3}-S_{2}\right)

8. If S_{2} S_{3} S_{5} are the sums of 2 n 3 n 5 n terms of an A.P. show that S_{5}=5\left(S_{3}-S_{2}\right)

Q.35 The common ratio of a geometric sequence cannot be(a) 3(b) 1(c) 2(d) 0
Q.35 The common ratio of a geometric sequence cannot be(a) 3(b) 1(c) 2(d) 0

Q.35 The common ratio of a geometric sequence cannot be(a) 3(b) 1(c) 2(d) 0

Sum the following series upto n terms.6. 2^{2}+5^{2}+8^{2}+\ldots
Sum the following series upto  n  terms.6.  2^{2}+5^{2}+8^{2}+\ldots

Sum the following series upto n terms.6. 2^{2}+5^{2}+8^{2}+\ldots

3. If the 5 th term of an A.P. is 16 and the 20 th term is 46 what is its 12 th term?
3. If the 5 th term of an A.P. is 16 and the 20 th term is 46  what is its 12 th term?

3. If the 5 th term of an A.P. is 16 and the 20 th term is 46 what is its 12 th term?

Q.53 A number H is said to be the harmonic mean between two numbers a and b if a H b form(a) H.P.(b) A.P.(c) G.P.(d) harmonic
Q.53 A number  H  is said to be the harmonic mean between two numbers  a  and  b  if  a H b  form(a) H.P.(b) A.P.(c) G.P.(d) harmonic
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Q.53 A number H is said to be the harmonic mean between two numbers a and b if a H b form(a) H.P.(b) A.P.(c) G.P.(d) harmonic

3. In the following questions three of the five elements a l d. n S of an A.P. are qiven. Find the missing elements.(ii) a=3 d=6 S_{n}=363
3. In the following questions three of the five elements  a l  d.  n S  of an A.P. are qiven. Find the missing elements.(ii)  a=3 d=6 S_{n}=363

3. In the following questions three of the five elements a l d. n S of an A.P. are qiven. Find the missing elements.(ii) a=3 d=6 S_{n}=363

12. If \frac{1}{a} \frac{1}{b} and \frac{1}{c} are in G.P. show that the common ratio is \pm \sqrt{\frac{a}{c}}
12. If  \frac{1}{a} \frac{1}{b}  and  \frac{1}{c}  are in G.P. show that the common ratio is  \pm \sqrt{\frac{a}{c}}

12. If \frac{1}{a} \frac{1}{b} and \frac{1}{c} are in G.P. show that the common ratio is \pm \sqrt{\frac{a}{c}}

9. If the n th term of the A.P. is 3 n-1 find the A.P.
9. If the  n th term of the A.P. is  3 n-1  find the A.P.

9. If the n th term of the A.P. is 3 n-1 find the A.P.

Sum the following series upto n terms.9. 2 \times 4 \times 7+3 \times 6 \times 10+4 \times 8 \times 13+\ldots
Sum the following series upto  n  terms.9.  2 \times 4 \times 7+3 \times 6 \times 10+4 \times 8 \times 13+\ldots

Sum the following series upto n terms.9. 2 \times 4 \times 7+3 \times 6 \times 10+4 \times 8 \times 13+\ldots

2. Sum the seriesi) -3+(-1)+1+3+5+\ldots .+a_{16}
2. Sum the seriesi) -3+(-1)+1+3+5+\ldots .+a_{16}

2. Sum the seriesi) -3+(-1)+1+3+5+\ldots .+a_{16}

11. If l m n are the p th q th and r th terms of an A.P. show thati) l(q-r)+m(r-p)+n(p-q)=0 ii) p(m-n)+q(n-l)+r(l-m)=0
11. If  l m n  are the  p  th  q  th and  r  th terms of an A.P. show thati)  l(q-r)+m(r-p)+n(p-q)=0 ii)  p(m-n)+q(n-l)+r(l-m)=0

11. If l m n are the p th q th and r th terms of an A.P. show thati) l(q-r)+m(r-p)+n(p-q)=0 ii) p(m-n)+q(n-l)+r(l-m)=0

6. Find vulgar fractions equivalent to the following recurring decimals.vi) 1.147
6. Find vulgar fractions equivalent to the following recurring decimals.vi)  1.147

6. Find vulgar fractions equivalent to the following recurring decimals.vi) 1.147

Example 1. If a b c are three positive numbers in H.P. prove that\[a^{2}+c^{2}>2 b^{2}\]
Example 1. If  a b c  are three positive numbers in H.P. prove that\[a^{2}+c^{2}>2 b^{2}\]

Example 1. If a b c are three positive numbers in H.P. prove that\[a^{2}+c^{2}>2 b^{2}\]

13. S_{7} and S_{9} are the sums of the first 7 and 9 terms of an A.P. respectively. If \frac{S_{9}}{S_{7}}=\frac{18}{11} and a_{7}=20 find the series.
13.  S_{7}  and  S_{9}  are the sums of the first 7 and 9 terms of an A.P. respectively. If  \frac{S_{9}}{S_{7}}=\frac{18}{11}  and  a_{7}=20  find the series.

13. S_{7} and S_{9} are the sums of the first 7 and 9 terms of an A.P. respectively. If \frac{S_{9}}{S_{7}}=\frac{18}{11} and a_{7}=20 find the series.

1. Write the first four terms of the following sequences ifii) a_{n}=(-1)^{n} n^{2}
1. Write the first four terms of the following sequences ifii)   a_{n}=(-1)^{n} n^{2}
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1. Write the first four terms of the following sequences ifii) a_{n}=(-1)^{n} n^{2}

1. Find the sum of each of the following A.P.s to the indicated number of terms.(暗) 17 \frac{1}{2} 14 \frac{1}{6} 10 \frac{5}{6} \ldots ; to 24 terms
1. Find the sum of each of the following A.P.s to the indicated number of terms.(暗)  17 \frac{1}{2} 14 \frac{1}{6} 10 \frac{5}{6} \ldots ;  to 24 terms
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1. Find the sum of each of the following A.P.s to the indicated number of terms.(暗) 17 \frac{1}{2} 14 \frac{1}{6} 10 \frac{5}{6} \ldots ; to 24 terms

Q.29 A_{1} A_{2} \ldots A_{n} are said to be n arithmetic means between a and b if a_{1} A_{1} A_{2} \ldots A_{n} b is(a) a sequence(b) not a sequence(c) G.P.(d) A.P.
Q.29  A_{1} A_{2} \ldots A_{n}  are said to be  n  arithmetic means between a and  b  if  a_{1} A_{1} A_{2} \ldots A_{n} b  is(a) a sequence(b) not a sequence(c) G.P.(d) A.P.
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Q.29 A_{1} A_{2} \ldots A_{n} are said to be n arithmetic means between a and b if a_{1} A_{1} A_{2} \ldots A_{n} b is(a) a sequence(b) not a sequence(c) G.P.(d) A.P.

1. Write the first four terms of the following arithmetic sequences if i) a_{1}=5 and other three consecutive terms are 232629
1. Write the first four terms of the following arithmetic sequences if i)  a_{1}=5  and other three consecutive terms are  232629
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1. Write the first four terms of the following arithmetic sequences if i) a_{1}=5 and other three consecutive terms are 232629

6. Find vulgar fractions equivalent to the following recurring decimals.iv) 1 . \dot{5} \dot{3}
6. Find vulgar fractions equivalent to the following recurring decimals.iv)  1 . \dot{5} \dot{3}
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6. Find vulgar fractions equivalent to the following recurring decimals.iv) 1 . \dot{5} \dot{3}

1. Sum the series.(b) 7+77+777+7777+\ldots to n terms
1. Sum the series.(b)  7+77+777+7777+\ldots  to  n  terms
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1. Sum the series.(b) 7+77+777+7777+\ldots to n terms

6. If S_{n}=n(2 n-1) then find the series.
6. If  S_{n}=n(2 n-1)  then find the series.
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6. If S_{n}=n(2 n-1) then find the series.

7. If a b c d are in G.P prove thatii) a^{2}-b^{2} b^{2}-c^{2} c^{2}-d^{2} are in G.P.
7. If  a b c d  are in G.P prove thatii)  a^{2}-b^{2} b^{2}-c^{2} c^{2}-d^{2}  are in G.P.
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7. If a b c d are in G.P prove thatii) a^{2}-b^{2} b^{2}-c^{2} c^{2}-d^{2} are in G.P.

5. Find the 18 th term of the A.P. if its 6 th term is 19 and the 9 th term is 31 .
5. Find the 18 th term of the A.P. if its 6 th term is 19 and the 9 th term is 31 .
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5. Find the 18 th term of the A.P. if its 6 th term is 19 and the 9 th term is 31 .

2. Find the first three terms of the sequence for each of the following general terms.(iv) \frac{(-1)^{n}}{n^{2}}
2. Find the first three terms of the sequence for each of the following general terms.(iv)  \frac{(-1)^{n}}{n^{2}}
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2. Find the first three terms of the sequence for each of the following general terms.(iv) \frac{(-1)^{n}}{n^{2}}

Example 3: Find the sum of the infinite G.P. 2 \sqrt{21} \ldots
Example 3: Find the sum of the infinite G.P.  2 \sqrt{21} \ldots
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Example 3: Find the sum of the infinite G.P. 2 \sqrt{21} \ldots

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