Students often refer to SSLC Maths Textbook Solutions and Class 10 Maths Chapter 3 Arithmetic Sequences and Algebra Important Extra Questions and Answers Kerala State Syllabus to clear their doubts.
SSLC Maths Chapter 3 Arithmetic Sequences and Algebra Important Questions and Answers
Arithmetic Sequences and Algebra Class 10 Extra Questions Kerala Syllabus
Arithmetic Sequences and Algebra Class 10 Kerala Syllabus Extra Questions
Question 1.
Algebraic form of an arithmetic sequence is xn = 3n +1. Common difference is
(a) 3
(b) 4
(c) -3
(d) 5
Answer:
(a) 3
Question 2.
Sum of the first n terms of an arithmetic sequence n2 + n. What is its common difference .
(a) 3
(b) 4
(c) 2
(d) 5
Answer:
(c) 2
Question 3.
Sum of the first n terms of an arithmetic sequence is n2 + n. What is its 13th term?
(a) 31
(b) 26
(c) 20
(d) 50
Answer:
(b) 26
Question 4.
a, a – 1, a – 2 ……. is an arithmetic sequence. What is its nth term?
(a) a + n + 1
(b) a + n – 1
(c) a – n – 1
(d) a – n + 1
Answer:
(d) a – n + 1
Question 5.
The algebraic form of the sequence \(\frac{1}{7}, \frac{3}{7}, \frac{5}{7}\) ……….?
(a) \(\frac{n}{7}\)
(b) \(\frac{2 n+1}{7}\)
(c) \(\frac{2 n-1}{7}\)
(d) None of these
Answer:
(c) \(\frac{2 n-1}{7}\)
Question 6.
Algebraic form of an arithmetic sequence is \(\frac{n+1}{3}\)
(a) What is its common difference ?
Answer:
\(\frac{1}{3}\)
(b) At what position 15 becomes a term of the sequence?
Answer:
44th term is 15
Question 7.
Sum of the first n terms of an arithmetic sequence is 3n2 + 2n
(a) What is its common difference?
(b) Find the sum of first 20 terms
Answer:
(a) 6
(b) 3 × 202 + 2 × 20 = 1240
Question 8.
\(\frac{1}{11}, \frac{2}{11}, \frac{3}{11} \ldots\) is an arithmetic sequence.
(a) Write the algebraic form of this sequence?
(b) Find the sum of first 10 terms of this sequence?
Answer:
(a) \(\frac{n}{11}\)
(b) \(\frac{1+2+3+\cdots 10}{11}=\frac{55}{11}\) = 5
Question 9.
nth term of an arithmetic sequence is 1 – 4n
(a) What is the common difference ?
(b) Find the sum of first 25 terms of this sequence.
Answer:
(a) – 4
(b) x25 = 1 – 4 x 25 = -99, x1 = 1 – 4 × 1 = -3
Sum =(x1 + x25) × \(\frac{25}{2}\)
= (-3 + -99) × \(\frac{25}{2}\) = -1275
Question 10.
Sum of the first«terms of an arithmetic sequence is 5«2 + 3n .
(a) What is the common difference ?
(b) Write the algebraic form of this sequence.
Answer:
(a) d = 10
(b) f = 8, d = 10
⇒ xn = 10n – 2
Question 11.
Consider the arithmetic sequence 1, 4, 7, 10….
(a) Write the nth term of the sequence?
(b) Find the expresson for the sum of first terms of this sequence.
(c) Calculate the sum of first 20 terms of this sequence.
Answer:
(a) xn = 3n – 2
(b) Sum = (x1 + xn) × \(\frac{n}{2}\) = (1 + 3n – 2) × \(\frac{n}{2}\)
= \(\frac{3 n^2}{2}-\frac{n}{2}\)
(c) \(\frac{3 \times 20^2}{2}-\frac{20}{2}\) = 590
Question 12.
Consider the arithmetic sequence 6, 10, 14,…
(a) What is the common difference of this sequence?
(b) Find the sum of first n terms of this sequence
(c) Can sum of some terms of this sequence 1225? How do you know this?
Answer:
(a) d = 4
(b) xn = 4n + 2
Sum =(6 + 4n + 2) × \(\frac{n}{2}\) = 2n2 +4 n
(c) All terms are even numbers. Sum of even numbers cannot be an odd number. 1225 cannot be the sum
Question 13.
First term of an arithmetic sequence is \(\frac{1}{2}\) and common difference \(\frac{3}{4}\)
(a) What is the algebra of this sequence?
(b) At what position 50 becomes a term of the se-quence
(c) What is the sum of first 11 terms?
Answer:
(a) xn = dn + (f – d)
= \(\frac{3}{4}\)n + \(\left(\frac{1}{2}-\frac{3}{4}\right)=\frac{3 n-1}{4}\)
(b) \(\frac{3 n-1}{4}\) = 50
⇒ 3n – 1 = 200,
3n = 201,
n = 67
(c) Sum = \(\frac{3(1+2+3+\cdots+11)-11}{4}=\frac{187}{4}\)
Question 14.
a + 1, a + 2, a + 3 ……….. is an arithmetic sequence
(a) What is its algebraic form?
(b) What is the sum of first 20 terms?
Answer:
(a) xn = a + n
(b) Sum = 20a + \(\frac{20 \times 21}{2}\)
= 20a + 210
Question 15.
Algebraic form ofan arithmetic sequence is 3n + 2
(a) What is the common difference?
(b) What is its 15th term? ‘
(c) What is the sum of first 29 terms?
Answer:
(a) d = 3
(b) x15 = 3 × 15 + 2 = 47
(c) Since x15th term is the middle term the first 29 terms sum will be x15 × 29
= 47 × 29
= 1363
Question 16.
Consider the arithmetic sequence -1, 3, 7, …………
(a) What is the common difference?
Answer:
d = 4
(b) Write the nth term of this sequence.
Answer:
4n – 5
(c) Is 95 a term of the sequence ?
Answer:
4n – 5 = 95
⇒ 4n = 100,
n = 25
(d) Calculate the sum of the terms upto 95.
Answer:
Sum = 4(1 + 2 + 3 + ……… + 25) – 5 × 25
= 4 × 325 – 125
= 1175
Question 17.
Sum of the first n terms of an arithmetic sequence is n2 +5n
(a) Find the sum of first 15 terms
Answer:
152 + 5 × 15 = 300
(b) WTiat is its 8th term?
Answer:
x8 = \(\frac{300}{15}\) = 20
(c) Write the algebraic form of the sequence.
Answer:
f = 6, d= 2
⇒ xn = 2n + 4
Question 18.
Sum of the first 3 terms of an arithmetic sequence is 15 . Sum of the first 4 terms is 28
(a) What is the 4th term?
Answer:
x4 = 28 – 15 = 13
(b) What is the sum of first 7 terms of the sequence?
Answer:
x4 × 7 = 91
(c) What is the third term ?
Answer:
2d = 13 – 5 = 8,
d = 4,
x3 = 13 – 4 = 9
(d) Express the sequence algebraically
Answer:
xn = 4n – 3
Question 19.
Sum of the first n terms of an arithmetic sequence is n2
(a) What is the first term?
Answer:
12 = 1
(b) Find the common difference
Answer:
12 + 22 = 5, x2 = 3, d = 2
(c) Write the sequence algebraically
Answer:
2n – 1
Question 20.
Sum of the first n terms of an arithmetic sequence is n2 + n
(a) Find the common difference of the arithmetic sequence.
Answer:
x1 = 12 + 1 = 2, x1 + x2 = 6
⇒ x2 = 4
d = 4 – 2 = 2
(b) Write the sequence argebraically
Answer:
xn = 2n
(c) Find the sum : 3 + 5 + 7 + …. + 51
Answer:
25th term is 50. Each term of 3, 5, 1…. is one more than 2, 4, 6….
3 + 5 + 7 + ……….. + 51 = 252 + 25 + 25 = 675
Question 21.
If \(\frac{1+3+5+\cdots(2 n-1)}{2+5+8+\cdots 23}\) = 9 then what is n ?
Answer:
\(\frac{n^2}{2+5+8+\cdots+23}\) = 9
2, 5, 8….23 has 8 terms. Sum is 100
\(\frac{n^2}{100}\) = 9
⇒ n2 = 900,
n = 30
Question 22.
nth term of an arithmetic sequence is 2n +1.
(a) Write the sequence in the numerical form
(b) What is its 100th term?
(c) Calculate the sum of first 100 terms
(d) Find the sum of first 100 terms of the arith-metic sequence 4, 6, 8
Answer:
(a) 3, 5, 7….
(b) 2 × 100+ = 201
(c) 2(1 +2 + 3 +….+100) + 100 = 10200
(d) The terms of 4, 6, 8,… are one more than the terms of 2n + 1 Sum is 10200 + 100 = 10300
Question 23.
Let x1, x2, x3 ……… be the terms of an arithmetic sequence.
Given that x1 + x5 + x10 + x15 + x20 + x24 = 225
Calculate the sum of first 24 terms of this sequence.
x1 + (x1 + 4d) + (x1 + 9d) + (x1 + 14d) + (x1 + 19d) + (x1 + 23d) = 225
Answer:
6x1 + 69d = 225, 2x1 + 23d = 75
Sum = (x1 + x24) × \(\frac{24}{2}\)
= (x1 + x1 + 23d) × 12
= (2x1 + 23d) × 12
= 75 × 12 = 900
Question 24.
Algebraic form of an arithmetic sequence is \(\frac{3}{7}\) n +1
(a) What is the common difference?
(b) Obtain the expression for the sum of first n terms of this sequence
(c) Find the sum of first 21 terms of the sequence.
Answer:
(a) \(\frac{3}{7}\)
(b) Sum = (x1 + xn) × \(\frac{n}{2}=\frac{3 n^2+17 n}{14}\)
(c) \(\frac{3 \times 21^2+17 \times 21}{14}\) = 120
Question 25.
-117, -114, -111… is an arithmetic sequence.
(a) What is the common difference ?
(b) Find the nth term of the sequence
(c) At what position 0 becomes a term of the se-quence?
(d) What is the sum of first 79 terms of this se-quence?
Answer:
(a) 3
(b) 3n – 120
(c) 3n – 120 = 0 × n = 40
(d) When we consider 79 terms, 40th term will be the middle term. It is 0
Sum of 79 terms is 0
Question 26.
The sum of first 10 terms is 230 . The sum of first 15 terms is 495.
(a) If pn2 + qn is the sum of first n terms then write a pair of equations
Answer:
100p + 10q = 230
⇒ 10p + q = 23
225 + 15q = 495
⇒ 15p + q = 33
5p = 10, p = 2, q = 3
(b) Find p and q and write the expression of the sum of first n terms
Answer:
2n2 +3n
(c) Find the algebraic form of the sequence
Answer:
xn = 4n +1
Question 27.
10 times 10th term of an arithmetic sequence is 20 times its 20th term.
(a) Find the 30th term of the sequence.
(b) What is the product of first 30 terms ?
Answer:
(a) 10(f + 9d) = 20(f + 19d)
⇒ f + 29d = 0, x30 = 0
(b) 0
Question 28.
The sums of the first n terms of three arithmetic sequences are y1, y2 and y3. The first term of each is 1 and common differences are 1,2 and 3 in the order. Prove that y1 + y3 = 2y2
Answer:
s1 = 1 + 2 + 3 … n = \(\frac{n(n+1)}{2}\)
s2 = 1 + 3 + 5 + …….. + 2n – 1 = n2
For the third sequence, f = 1, d = 3
nthth term is 3n – 2 Sum of first n terms = \(\frac{3 n^2}{2}-\frac{n}{2}\)
s1 + s3 = \(\frac{n(n+1)}{2}+\frac{3 n^2}{2}-\frac{n}{2}\)
On simplifying s1 + s3 = 2n2= 2s2
Question 29.
Sum of the first n odd numbers is k
(a) What is the sum of first n even numbers?
Answer:
k + √k
(b) What is the sum of first n natural numbers ?
Answer:
\(\frac{k+\sqrt{k}}{2}\)
(c) Find \(\frac{1+2+3+4+\cdots+15}{16+17+18+\cdots+30}\)
Answer:
Find \(\frac{8}{23}\)
Question 30.
Consider the sequence of numbers which leaves the remainder 4 on dividing by 7 .
(a) Write the algebraic form of the sum of the terms of this sequence.
(b) Calculate the sum of first 20 terms of this se-quence
Answer:
(a) 4, 11, 18…. is the sequence
Sum of the first n terms = pn2 + qn
p + q = 4, 4p + 2q = 15
Solving q = \(\frac{1}{2}\), p = \(\frac{7}{2}\)
Sum of first n terms = \(\frac{7}{2}\) n2 + \(\frac{1}{2}\)n
(b) \(\frac{7}{2}\) × 202 + \(\frac{20}{2}\)
= 1400 + 10
= 1410