CBSE Class 10 Mathematics Arithmetic Progressions Worksheet Set 06

Fill in the Blanks

DIRECTIONS : Complete the following statements with an appropriate word / term to be filled in the blank space(s).

Question. 4, 10, 16, 22, ........, .......... .
Answer: 28, 34

Question. 1, –1, –3, – 5, ....... , ......... .
Answer: –7, –9

Question. 11^th term from last term of an A.P. 10, 7, 4......... , – 62, is ......... .
Answer: –32

Question. In a flower bed, there are 23 rose plants in the first row, 21 in the second, 19 in the third, and so on. There are 5 rose plants in the last row. Number of rows in the flower bed is ............... .
Answer: n = 10

Question. Sum of 1 + 3 + 5 + .... + 1999 is ......... .
Answer: 1000/2[2(1) (1000 1)2]

Question. The sum of 8 A.Ms between 3 and 15 is ................... .
Answer: 72 [ 8 (3 + 15 / 2 ) etc.]

Question. The sum of n terms of an A.P. is 4n² – n. The common  difference = .................. .
Answer: 11 [S₂ = 4(2)² – 2 ⇒ 14
S₁ = 4(1)² – 1 ⇒ 3 etc.]

Question. The difference of corresponding terms of two A.P’s will be .................... .
Answer: another A.P.

Question. Sum of all the integers between 100 and 1000 which are divisible by 7 is ...................
Answer: 70336 [Hint : S = 105 + 112 + ... 994 and 105 + (n – 1)7 = 994 ⇒ 105 + 7n – 7 = 994 ⇒ n = 128 etc.]

True / False

DIRECTIONS : Read the following statements and write your answer as true or false.

Question. In an AP with first term a and common difference d, the n^th term (or the general term) is given by a_n = a + (n – 1)d.
Answer: True

Question. The balance money ( in `) after paying 5% of the total loan of ` 1000 every month is 950, 900, 850, 800, . . . 50. represented A.P.
Answer: True

Question. 2, 4, 8, 16, ............. is not an A.P.
Answer: True

Question. 10^th term of A.P. 2, 7, 12, ......... is 45.
Answer: False

Question. 301 is a term of A.P. 5, 11, 17, 23, ............. .
Answer: False

Question. The general form of an A.P. is a, a + d, a + 2d, a + 3d, .............
Answer: True

Question. In an Arithmetic progression, the first term is denoted by ‘a’ and ‘d ’ is called the common difference.
Answer: True

Question. If a_n+1 – a_n = same for all ‘n’, then the given numbers form an A.P.
Answer: True

Question. If S_n of A.P. is 3n² + 2n, then the first term of A.P. is 3.
Answer: False

VERY SHORT ANSWER QUESTIONS(1 MARK)

Question. If the sum of first m terms of an AP is am2 + bm, find the common difference.
Answer : S_m= am² + bm, S₁= a+b= a₁, S₂ = 4a+2b= a₁+ a₂
a₂ = S₂- S₁= 4a+2b- (a+b) = 3a +b , d= a₂-a₁= 3a+b-(a+b)= 2a

Question. Three numbers are in AP and their sum is 24. Find the middle term.
Answer : Let the three numbers of the AP be a-d, a, a+d. So a-d +a+a+d= 24 ⟹ 3a=24,a=24/3=8. Hence the middle term =8.

Question. Find the sum of first 8 multiples of 3.
Answer : First 8 multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24 These numbers are in A.P.
where a = 3, d = 3 and n = 8 , a_n=24, S_n= 𝑛/2(a 1+ a_n), S₈= 8/2(3+ 24)= 4x27=108

Question. Find the sum: 34+ 32+30+……+10
Answer : Given, 34 + 32 + 30 + … + 10, first term, a = 34, d = a2−a1 = 32−34 = −2, Let 10 be the nth term of this A.P., an= a +(n−1)d, 10 = 34+(n−1)(−2), −24 = (n −1)(−2), 12 = n −1, n = 13, S_n = n/2 (a +l) , l = 10, S_n = n/2 (34 + 10) = 13/2 x44= 286

Question. Find the number of terms of an AP 5, 9,13, …, 185.
Answer : a₁=5, d=9-5=4, a_n=185, 185 = 5 + (n - 1)4, 185 = 5 + 4n -4,185 = 1 + 4n, 185 - 1 = 4n, 184 = 4n, n= 184/4 = 46

Question. Find the number of natural numbers between 101 and 999 which are divisible by both 2 and 5
Answer : Since, the number is divisible by both 2 and 5, means it must be divisible by 10.AP = 110, 120, 130,..., 990, a = 110, d = 10, nth term of the AP = 990
a+(n-1)d=990, 110+(n-1)10=990, (n-1)10=990-110,(n-1)= 880/10, n-1=88, n=88+1, n=89

Question. The fourth term of an AP is 11 and the eleventh term is 25. Determine the first term and common difference.
Answer : a + 3d = 11….(1) a + 10d = 25…..(2)
Subtracting equation (1) from equation (2 )
a + 10d – (a + 3d) = 25 – 11, 7d = 14, d = 2,
Putting value of d=2 in the equation 2,
a + 10x2 = 25, a + 20 = 25, a = 25 – 20, a = 5

Question. In an AP a = 15, d = -3, an = 0, then find the value of n.
Answer : First term (a) =15,Common difference (d) = -3,
Last term(an) = 0, 0 = 15 + (n - 1)-3 -15 = -3n + 3, -15 - 3 = -3n, -18 = -3n, n=6