Which of the following are APs ? If they form an AP, find the common difference d and write three more terms. (i) 2, 4, 8, 16, .... (ii) 2, (iii) -1.2, -3.2, -5.2, -7.2, ........ (iv) -10, -6, -2, 2, ......... (v) 3, 3 + , .......... (vi) 0.2, 0.22, 0.222, 0.2222, .......... (vii) 0, -4, -8, -12, ........ (viii) (ix) 1, 3, 9, 27, ........ (x) a, 2a, 3a, 4a, .......... (xi) a, a², a³, a⁴, ........... (xii) ........... (xiii) ........... (xiv) 1², 3², 5², 7², ...... (xv) 1², 5², 7², 73, ......

Question

KnowledgeBoat · Accepted Answer

(i) Given, 2, 4, 8, 16, ... ⇒ a₂ - a₁ = 4 - 2 = 2, ⇒ a₃ - a₂ = 8 - 4 = 4, Since, a₃ - a₂ a₂ - a₁ Hence, the given list of numbers do not form an A.P. (ii) Given, 2, ⇒ a₂ - a₁ = , ⇒ a₃ - a₂ = , ⇒ a₄ - a₃ = . i.e., a_{k + 1} - a_k is same every time. So, the given list of numbers forms A.P. with common difference = . Next three terms are : ⇒ = 4, ⇒ 4 + . ⇒ = 5. Hence, for the given A.P. common difference = and next three terms are 4, and 5. (iii) Given, -1.2, -3.2, -5.2, -7.2, .... ⇒ a₂ - a₁ = -3.2 - (-1.2) = -3.2 + 1.2 = -2, ⇒ a₃ - a₂ = -5.2 - (-3.2) = -5.2 + 3.2 = -2, ⇒ a₄ - a₃ = -7.2 - (-5.2) = -7.2 + 5.2 = -2. i.e., a_{k + 1} - a_k is same every time. So, the given list of numbers forms A.P. with common difference = -2. Next three terms are : ⇒ -7.2 + (-2) = -7.2 - 2 = -9.2, ⇒ -9.2 + (-2) = -9.2 - 2 = -11.2 ⇒ -11.2 + (-2) = -11.2 - 2 = -13.2 Hence, for the given A.P. common difference = -2 and next three terms are -9.2, -11.2 and -13.2 (iv) Given, -10, -6, -2, 2, ......... ⇒ a₂ - a₁ = -6 - (-10) = -6 + 10 = 4, ⇒ a₃ - a₂ = -2 - (-6) = -2 + 6 = 4, ⇒ a₄ - a₃ = 2 - (-2) = 2 + 2 = 4. i.e., a_{k + 1} - a_k is same every time. So, the given list of numbers forms A.P. with common difference = 4. Next three terms are : ⇒ 2 + 4 = 6, ⇒ 6 + 4 = 10, ⇒ 10 + 4 = 14. Hence, for the given A.P. common difference = 4 and next three terms are 6, 10 and 14 (v) Given, 3, 3 + , .......... ⇒ a₂ - a₁ = , ⇒ a₃ - a₂ = , ⇒ a₄ - a₃ = . i.e., a_{k + 1} - a_k is same every time. So, the given list of numbers forms A.P. with common difference = . Next three terms are : ⇒ , ⇒ , ⇒ . Hence, for the given A.P. common difference = and next three terms are . (vi) Given, 0.2, 0.22, 0.222, 0.2222, ........ ⇒ a₂ - a₁ = 0.22 - 0.2 = 0.02, ⇒ a₃ - a₂ = 0.222 - 0.22 = 0.002. Since, a₃ - a₂ a₂ - a₁ Hence, the given list of numbers do not form an A.P. (vii) Given, 0, -4, -8, -12, ........ ⇒ a₂ - a₁ = -4 - 0 = -4, ⇒ a₃ - a₂ = -8 - (-4) = -8 + 4 = -4, ⇒ a₄ - a₃ = -12 - (-8) = -12 + 8 = -4. i.e., a_{k + 1} - a_k is same every time. So, the given list of numbers forms A.P. with common difference = -4. Next three terms are : ⇒ -12 + (-4) = -12 - 4 = -16, ⇒ -16 + (-4) = -16 - 4 = -20, ⇒ -20 + (-4) = -20 - 4 = -24. Hence, for the given A.P. common difference = -4 and next three terms are -16, -20 and -24. (viii) Given, ⇒ a₂ - a₁ = = 0, ⇒ a₃ - a₂ = = 0, ⇒ a₄ - a₃ = = 0, i.e., a_{k + 1} - a_k is same every time. So, the given list of numbers forms A.P. with common difference = 0. Next three terms are : , , . Hence, for the given A.P. common difference = 0 and next three terms are . (ix) Given, 1, 3, 9, 27, ......... ⇒ a₂ - a₁ = 3 - 1 = 2, ⇒ a₃ - a₂ = 9 - 3 = 6. Since, a₃ - a₂ a₂ - a₁ Hence, the given list of numbers do not form an A.P. (x) Given, a, 2a, 3a, 4a, ..... ⇒ a₂ - a₁ = 2a - a = a, ⇒ a₃ - a₂ = 3a - 2a = a, ⇒ a₄ - a₃ = 4a - 3a = a. i.e., a_{k + 1} - a_k is same every time. So, the given list of numbers forms A.P. with common difference = a. Next three terms are : ⇒ 4a + a = 5a, ⇒ 5a + a = 6a, ⇒ 6a + a = 7a. Hence, for the given A.P. common difference = a and next three terms are 5a, 6a and 7a. (xi) Given, a, a², a³, a⁴, ......... ⇒ a₂ - a₁ = a² - a = a(a - 1), ⇒ a₃ - a₂ = a³ - a² = a²(a - 1). Since, a₃ - a₂ a₂ - a₁ Hence, the given list of numbers do not form an A.P. (xii) Given, ⇒ ........... ⇒ , ........... ⇒ a₂ - a₁ = , ⇒ a₃ - a₂ = , ⇒ a₄ - a₃ = . i.e., a_{k + 1} - a_k is same every time. So, the given list of numbers forms A.P. with common difference = . Next three terms are : ⇒ , ⇒ , ⇒ . Hence, for the given A.P. common difference = and next three terms are . (xiii) Given, ........... ⇒ a₂ - a₁ = , ⇒ a₃ - a₂ = . Since, a₃ - a₂ a₂ - a₁ Hence, the given list of numbers do not form an A.P. (xiv) Given, ⇒ 1², 3², 5², 7², ...... ⇒ 1, 9, 25, 49, ......... ⇒ a₂ - a₁ = 9 - 1 = 8, ⇒ a₃ - a₂ = 25 - 9 = 16. Since, a₃ - a₂ a₂ - a₁ Hence, the given list of numbers do not form an A.P. (xv) Given, ⇒ 1², 5², 7², 73, ...... ⇒ 1, 25, 49, 73, ......... ⇒ a₂ - a₁ = 25 - 1 = 24, ⇒ a₃ - a₂ = 49 - 25 = 24, ⇒ a₄ - a₃ = 73 - 49 = 24. i.e., a_{k + 1} - a_k is same every time. So, the given list of numbers forms A.P. with common difference = 24. Next three terms are : ⇒ 73 + 24 = 97, ⇒ 97 + 24 = 121, ⇒ 121 + 24 = 145. Hence, for the given A.P. common difference = 24 and next three terms are 97, 121 and 145.

S. No.	a	d	n	a_n
(i)	7	3	8	-----
(ii)	-18	----	10	0
(iii)	----	-3	18	-5
(iv)	-18.9	2.5	------	3.6
(v)	3.5	0	105	--------

Mathematics

AP

Related Questions

Fill in the blanks in the following table, given that a is the first term, d the common difference and a_n the nth term of the AP :
S.
No. a d n a_n
(i) 7 3 8 -----
(ii) -18 ---- 10 0
(iii) ---- -3 18 -5
(iv) -18.9 2.5 ------ 3.6
(v) 3.5 0 105 --------

Choose the correct choice in the following and justify :
30th term of the A.P. : 10, 7, 4, ….., is
97
77
-77
-87

Mathematics

AP

Related Questions

Fill in the blanks in the following table, given that a is the first term, d the common difference and an the nth term of the AP :S.No.adnan(i)738-----(ii)-18----100(iii)-----318-5(iv)-18.92.5------3.6(v)3.50105--------

Choose the correct choice in the following and justify :30th term of the A.P. : 10, 7, 4, ….., is9777-77-87

Fill in the blanks in the following table, given that a is the first term, d the common difference and a_n the nth term of the AP :
S.
No. a d n a_n
(i) 7 3 8 -----
(ii) -18 ---- 10 0
(iii) ---- -3 18 -5
(iv) -18.9 2.5 ------ 3.6
(v) 3.5 0 105 --------

Choose the correct choice in the following and justify :
30th term of the A.P. : 10, 7, 4, ….., is
97
77
-77
-87