**SEQUENCES AND SERIES**

**1.** The ratio of the sums of n terms of two A.P is (7n +1) : (4n +27).find the ratio of their 11^{th} terms.

**2.** the ratio of the sums of n terms of two A.P is (3n +4): (5n+6 ).find the ratio of their 5 th terms.

**3.** There are n A.M between 1 and 23 such thart the ratio of last mean to the first mean is 7:1. find the value of n.

**4.** AM.s have been inserted between 1 and 31 so that the ratio of 7th and (n-1)th means is 5:9. find the value of n

**5.** For what value of n, is the A M of a and b.

**6.** IF a , b, c are in AP . prove that b+c , c+a, a+b are in AP.

**7.** IF a , b, c are in AP. Prove that 1/bc, 1/ac,1/ab are also in AP.

**8.** IF a , b, c are in AP.prove that 1/( √b + √c),1/ (√c +√a),1/( √a + √b) are in AP.

**9.** IF a , b, c are in AP.prove that a^{2}(b +c), b^{2}(c +a), c^{2} (a+b) are in AP.

**10.** IF a , b, c are in AP. Prove that (b+c)^{2}- a ^{2},(c+a)^{2} -b^{2},(a+b)^{2}-c^{2} are in AP.

**11.** IF a , b, c are in AP .prove that a(b+c)/bc, b(c+a)/ca, c(a+b)/ab are in AP.

**12.** IF a , b, c are in AP prove that b+c-a, c+a-b,a+b-c .are in AP.

**13.** IF a , b, c are in AP.prove that bc-a^{2},ca-b^{2},ab-c^{2} are in AP.

**14.** IF a^{2},b^{2},c^{2} are in AP.then show that 1/(b+c),1/(c+a),1/(a+b) are in AP.

**15.** IF a^{2},b^{2},c^{2} are in AP.then show that a/b+c,b/c+a,c/a+b.are in AP.

**16.** IF (b+c-a)/a,(c+a-b)/b,(a+b-c)/c are in AP.prove that 1/a,1/b,1/c are also in AP.

**17.** IF a^{2}(b+c),b^{2}(c+a),c^{2}(a+b) are in AP.Show that either a,b,c are in AP. Or ab+bc+ac=0

**18.** IF 1/x+y, 1/2y, 1/y+z are in AP.Prove that x, y, z are in GP.

**19.** If a,b,c are in GP.and then show that x, y, z are in AP

**20.** If p,q,r are in AP.show that pth,qth,and rth terms of any GP are in GP.

**21.** Sum the following series up to n terms 9 + 99+ 999 + ..............

**22.** Sum the following series up to n terms 7 + 77 + 777 + ..................

**23.** Sum the following series up to n terms 0.3 + 0.33 + 0.333 + ..................

**24.** One side of a square is 10cm. The mid points of its sides are joined to form another square whose midpoints are again joined to form one more square and this process is continued indefinitely.find the sum of the areas of all the squares so formed.

**25.** Find two numbers whose AM=34 and GM=16

**26.** The AM of two positive numbers a and b (a>b)is twice the GM provethat a:b =2 +√3 : 2-√3

**27.** For the what value of n ,(a ^{n-1} + b ^{n-1}) / (a ^{n }+b ^{n} )is GM of a and b .?

**28.** Construct a quadriatic equation in x so that AM of its root is Aand GM is G.

**29.** Prove that product of n GM.s between a and b.is ( √ab)^{n}.

**30.** Find n if (a^{n}+b^{n})/(a ^{n-1} + b ^{n-1}) is GM of a and b.

**31.** If b is the GM between a and c P.T (a^{2} + b^{2}) (b^{2} + c^{2}) = (ab + bc) ^{2}

**32.** If a,b,c are in AP ,P.T x, x, x are in GP.

**33.** Find Sn of x(x + y) + x^{2}(x^{2} + y^{2}) + x^{3}(x^{3} + y^{3}) + ..............

**34.** Find Sn of 1+(1 + b)r + (1 + b + b^{2})r^{2} +(1 + b + b^{2} + b^{3})r^{3} + ..............

**35.** Find Sn (x + y) + (x^{2} + xy + y^{2}) + (x^{3} + x^{2}y + xy^{2 }+ y^{3}) + .............

**36.** A ball is dropped from a height of 48 m and rebounds two- third of the distance it falls .if it continues to fall and rebounds in this way how far it travels before coming to rest.

**37.** One side of an equilateral triangle is 12 m .the midpoints of its sides are joined to form another triangle whose midpoints in turn are joined to form still another triangle. This process is continued till 6 triangles are formed.Find sum of perimeters of all the triangles.

**38.** The sum of 4 numbers in GP is 40 and arithemetic mean of first and last is 18 .find the numbers.

**39.** The AMof 2 numbers exceeds theGM by 8 and ratio of the numbers is 4 .find the numbers.nb

**40.** If**a**n=n^{2}+3^{n} .find Sn.

Find the sum of the following series up to n terms.

**55.** Find the sum of the series 12 + 42 + 72 +….

**56.** 1.22 + 2.32 + 3.42 + ......