QuestionsQuestion 1Find the common difference and write the next two terms of each arithmetic sequence.(i)9,16,23,…9, 16, 23, \ldots9,16,23,…(ii)5,5+2,5+22,…5, 5+\sqrt{2}, 5+2\sqrt{2}, \ldots5,5+2,5+22,…SolutionTheoryQuestion 2Write the first three terms of each arithmetic sequence, with given information.(i)a1=2,d=13a_1=2, d=13a1=2,d=13(ii)a1=12,d=−13a_1=12, d=-13a1=12,d=−13SolutionTheoryQuestion 3Find an+1a_{n+1}an+1 and a2na_{2n}a2n if an=4+3na_n = 4 + 3nan=4+3n.SolutionTheoryQuestion 4Find the indicated term of each of the following arithmetic sequences:(i)a1=3,d=7,a14a_1=3, d=7, a_{14}a1=3,d=7,a14(ii)8,3,−2,…,a128, 3, -2, \ldots, a_{12}8,3,−2,…,a12SolutionTheoryQuestion 5The 18th and 30th terms of an arithmetic sequence are 367 and 499 respectively. How many terms of this sequence are less than 1000?SolutionTheoryQuestion 6Is 301 a term of the A.P. 5,11,17,…5, 11, 17, \ldots5,11,17,…?SolutionTheoryQuestion 7If 2x,x+8,3x+12x, x+8, 3x+12x,x+8,3x+1 are in A.P., then find the value of xxx.SolutionTheoryQuestion 8Which term of the A.P. 3,8,13,…3, 8, 13, \ldots3,8,13,… is 123?SolutionTheoryQuestion 9Which term of the A.P. 30,29.5,29,…30, 29.5, 29, \ldots30,29.5,29,… is the first negative term?SolutionTheoryQuestion 10The 7th and 21st terms of an A.P. are 37 and 107 respectively. Find the A.P. and its 100th term.SolutionTheoryQuestion 11If 1a−c,1b−c,1b−a\dfrac{1}{a-c}, \dfrac{1}{b-c}, \dfrac{1}{b-a}a−c1,b−c1,b−a1 are in A.P., then show that a−ba−c=a−cb−a\dfrac{a-b}{a-c}=\dfrac{a-c}{b-a}a−ca−b=b−aa−c.SolutionTheoryQuestion 12How many numbers of three digits are divisible by 7?SolutionTheoryQuestion 13Find the 8th term from the end of the A.P. 8,11,14,…,1858, 11, 14, \ldots, 1858,11,14,…,185.SolutionTheoryQuestion 14Find the nthn^{th}nth term of the progression (37)10,(107)10,(177)10,…\left(\dfrac{3}{7}\right)^{10}, \left(\dfrac{10}{7}\right)^{10}, \left(\dfrac{17}{7}\right)^{10}, \ldots(73)10,(710)10,(717)10,…. Is the progression an A.P.?SolutionTheoryQuestion 15If the arithmetic progressions 3,10,17,…3, 10, 17, \ldots3,10,17,… and 63,65,67,…63, 65, 67, \ldots63,65,67,… are such that their nthn^{th}nth terms are equal, then find the value of nnn.SolutionTheoryQuestion 16If the pthp^{th}pth term of an A.P. is qqq and the qthq^{th}qth term is ppp, prove that its nthn^{th}nth term is (p+q−n)(p+q-n)(p+q−n).SolutionTheoryQuestion 17If 1a,1b,1c\dfrac{1}{a}, \dfrac{1}{b}, \dfrac{1}{c}a1,b1,c1 are in A.P., show that b=2aca+cb=\dfrac{2ac}{a+c}b=a+c2ac.SolutionTheoryQuestion 18If 1a,1b,1c\dfrac{1}{a}, \dfrac{1}{b}, \dfrac{1}{c}a1,b1,c1 are in A.P., show that the common difference is a−c2ac\dfrac{a-c}{2ac}2aca−c.SolutionTheoryQuestion 19If aka_kak and ama_mam denotes two different terms of an A.P., show that its nthn^{th}nth term is ak+(n−k)(ak−amk−m)a_k+(n-k)\left(\dfrac{a_k-a_m}{k-m}\right)ak+(n−k)(k−mak−am).SolutionTheoryQuestion 20If a1,a2,a3,…,ana_1, a_2, a_3, \ldots, a_na1,a2,a3,…,an are positive and in A.P., prove that 1a1+a2+1a2+a3+⋯+1an−1+an=n−1a1+an\dfrac{1}{\sqrt{a_1}+\sqrt{a_2}}+\dfrac{1}{\sqrt{a_2}+\sqrt{a_3}}+\cdots+\dfrac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}=\dfrac{n-1}{\sqrt{a_1}+\sqrt{a_n}}a1+a21+a2+a31+⋯+an−1+an1=a1+ann−1.SolutionTheoryQuestion 21If the roots of the equation (b−c)x2+(c−a)x+(a−b)=0(b-c)x^2+(c-a)x+(a-b)=0(b−c)x2+(c−a)x+(a−b)=0 are equal, show that a,b,ca, b, ca,b,c are in A.P.SolutionTheoryQuestion 22If the sides of a right-angled triangle are in A.P., find the ratio of its sides.SolutionTheoryQuestion 23If the nthn^{th}nth term of a progression is a linear expression in nnn, then prove that this progression is an A.P.SolutionTheory