# 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}

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?

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)

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

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. 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}}

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

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

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.

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

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

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

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 .

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