**Question 1. Find the sum of the following APs.**

**(i) 2, 7, 12,…., to 10 terms.**

**(ii) − 37, − 33, − 29,…, to 12 terms**

**(iii) 0.6, 1.7, 2.8,…….., to 100 terms**

**(iv) 1/15, 1/12, 1/10, ……, to 11 terms**

**Solution:**

(i)Given, 2, 7, 12,…, to 10 termsFor this A.P., we have,

first term, a = 2

common difference, d = a2 − a1 = 7−2 = 5

no. of terms, n = 10

Sum of nth term in AP series is,

S

_{n}= n/2 [2a +(n-1)d]Substituting the values,

S

_{10}= 10/2 [2(2)+(10 -1)×5]= 5[4+(9)×(5)]

= 5 × 49 = 245

(ii)Given, −37, −33, −29,…, to 12 termsFor this A.P.,we have,

first term, a = −37

common difference, d = a2− a1

= (−33)−(−37)

= − 33 + 37 = 4

no. of terms, n = 12

Sum of nth term in AP series is,

S

_{n}= n/2 [2a+(n-1)d]Substituting the values,

S

_{12}= 12/2 [2(-37)+(12-1)×4]= 6[-74+11×4]

= 6[-74+44]

= 6(-30) = -180

(iii)Given, 0.6, 1.7, 2.8,…, to 100 termsFor this A.P.,

first term, a = 0.6

common difference, d = a2 − a1 = 1.7 − 0.6 = 1.1

no. of terms, n = 100

Sum of nth term in AP series is,

Sn = n/2[2a +(n-1)d]

S12 = 50/2 [1.2+(99)×1.1]

= 50[1.2+108.9]

= 50[110.1]

= 5505

(iv)Given, 1/15, 1/12, 1/10, ……, to 11 termsFor this A.P.,

first term, a = 1/5

common difference, d = a2 –a1 = (1/12)-(1/5) = 1/60

number of terms, n = 11

Sum of nth term in AP series is,

Sn = n/2 [2a + (n – 1) d]

Substituting the values, we have,

= 11/2(2/15 + 10/60)

= 11/2 (9/30)

= 33/20

**Question 2. Find the sums given below:**

**(i) 7+ 10 1/2 + 14 + ……… + 84**

**(ii) 34 + 32 + 30 + ……….. + 10**

**(iii) − 5 + (− 8) + (− 11) + ………… + (− 230)**

**Solutions:**

(i)Given,First term, a = 7

nth term, a

_{n }= 84Common difference, d = 10 1/2 – 7 = 21/2 – 7 = 7/2

Let 84 be the nth term of this A.P.

Then,

a

_{n}= a(n-1)dSubstituting these values,

84 = 7+(n – 1)×7/2

77 = (n-1)×7/2

22 = n−1

n = 23

We know that, sum of n term is;

S

_{n}= n/2 (a + l), l = 84S

_{n }= 23/2 (7+84)= (23×91/2) = 2093/2

= 1046 1/2

(ii)Given,first term, a = 34

common difference, d = a

_{2}−a_{1}= 32−34 = −2nth term, an= 10

Let 10 be the nth term of this A.P.,

Now,

a

_{n }= a +(n−1)d10 = 34+(n−1)(−2)

−24 = (n −1)(−2)

12 = n −1

n = 13

Sum of n terms is;

S

_{n }= n/2 (a +l), l = 10= 13/2 (34 + 10)

= (13×44/2) = 13 × 22

= 286

(iii)Given:First term, a = −5

nth term, a

_{n}= −230Common difference, d = a

_{2}−a_{1}= (−8)−(−5)⇒d = − 8+5 = −3

Let us assume −230 be the nth term of this A.P.

Since,

a

_{n}= a+(n−1)d−230 = − 5+(n−1)(−3)

−225 = (n−1)(−3)

(n−1) = 75

n = 76

Sum of n terms, is equivalent to,

S

_{n}= n/2 (a + l)= 76/2 [(-5) + (-230)]

= 38(-235)

= -8930

**Question 3. In an AP**

**(i) Given a = 5, d = 3, an = 50, find n and Sn.**

**(ii) Given a = 7, a13 = 35, find d and S13.**

**(iii) Given a12 = 37, d = 3, find a and S12.**

**(iv) Given a3 = 15, S10 = 125, find d and a10.**

**(v) Given d = 5, S9 = 75, find a and a9.**

**(vi) Given a = 2, d = 8, Sn = 90, find n and an.**

**(vii) Given a = 8, an = 62, Sn = 210, find n and d.**

**(viii) Given an = 4, d = 2, Sn = − 14, find n and a.**

**(ix) Given a = 3, n = 8, S = 192, find d.**

**(x) Given l = 28, S = 144 and there are total 9 terms. Find a.**

**Solutions:**

(i)Given values, we have,a = 5, d = 3, a

_{n}= 50The nth term in an AP,

a

_{n }= a +(n −1)d,Substituting the given values, we have,

⇒ 50 = 5+(n -1)×3

⇒ 3(n -1) = 45

⇒ n -1 = 15

Obtaining the value of n, we get,

⇒ n = 16

Now, sum of n terms is equivalent to,

S

_{n}= n/2 (a +an)S

_{n}= 16/2 (5 + 50) = 440

(ii)Given values, we have,a = 7, a

_{13}= 35The nth term in an AP,

a

_{n}= a+(n−1)d,Substituting the given values, we have,

⇒ 35 = 7+(13-1)d

⇒ 12d = 28

⇒ d = 28/12 = 2.33

Now, S

_{n}= n/2 (a+an)Obtaining the final value, we get,

S

_{13}= 13/2 (7+35) = 273

(iii)Given values, we have,a

_{12}= 37, d = 3The nth term in an AP,

a

_{n}= a+(n −1)d,Substituting the given values, we have,

⇒ a

_{12}= a+(12−1)3⇒ 37 = a+33

Obtaining the value of a, we get,

⇒ a = 4

Now, sum of nth term,

S

_{n}= n/2 (a+an)= 12/2 (4+37)

Obtaining the final value,

= 246

(iv)Given that,a3 = 15, S10 = 125

The formula of the nth term in an AP,

a

_{n}= a +(n−1)d,Substituting the given values, we have,

a

_{3}= a+(3−1)d15 = a+2d ………….. (i)

Also,

Sum of the nth term,

S

_{n}= n/2 [2a+(n-1)d]S

_{10}= 10/2 [2a+(10-1)d]125 = 5(2a+9d)

25 = 2a+9d …………….. (ii)

Solving eq (i) by (ii),

30 = 2a+4d ………. (iii)

And, by subtracting equation (iii) from (ii), we get,

−5 = 5d

that is,

d = −1

Substituting in equation (i),

15 = a+2(−1)

15 = a−2

a = 17 =

And,

a

_{10 }= a+(10−1)da

_{10}= 17+(9)(−1)a

_{10}= 17−9 = 8

(v)Given:d = 5, S

_{9}= 75Sum of n terms in AP is,

S

_{n}= n/2 [2a +(n -1)d]Substituting values, we get,

S

_{9}= 9/2 [2a +(9-1)5]25 = 3(a+20)

25 = 3a+60

3a = 25−60

a = -35/3

Also,

a

_{n}= a+(n−1)dSubstituting values, we get,

a

_{9}= a+(9−1)(5)= -35/3+8(5)

= -35/3+40

= (35+120/3) = 85/3

(vi)Given:a = 2, d = 8, S

_{n}= 90Sum of n terms in an AP is,

S

_{n}= n/2 [2a +(n -1)d]Substituting values, we get,

90 = n/2 [2a +(n -1)d]

⇒ 180 = n(4+8n -8) = n(8n-4) = 8n

^{2}-4nSolving the eq, we get,

⇒ 8n

^{2}-4n –180 = 0⇒ 2n

^{2}–n-45 = 0⇒ 2n

^{2}-10n+9n-45 = 0⇒ 2n(n -5)+9(n -5) = 0

⇒ (n-5)(2n+9) = 0

Since, n can only be a positive integer,

Therefore,

n = 5

Now,

∴ a

_{5}= 8+5×4 = 34

(vii)Given:a = 8, an = 62, S

_{n}= 210Since, sum of n terms in an AP is equivalent to,

S

_{n }= n/2 (a + an)210 = n/2 (8 +62)

Solving,

⇒ 35n = 210

⇒ n = 210/35 = 6

Now, 62 = 8+5d

⇒ 5d = 62-8 = 54

⇒ d = 54/5 = 10.8

(viii)Given :nth term, an = 4, common difference, d = 2, sum of n terms, S

_{n}= −14.Formula of the nth term in an AP,

a

_{n}= a+(n −1)d,Substituting the values, we get,

4 = a+(n −1)2

4 = a+2n−2

a+2n = 6

a = 6 − 2n …………………. (i)

Sum of n terms is;

S

_{n}= n/2 (a+an)-14 = n/2 (a+4)

−28 = n (a+4)

From equation (i), we get,

−28 = n (6 −2n +4)

−28 = n (− 2n +10)

−28 = − 2n

^{2}+10n2n

^{2}−10n − 28 = 0n

^{2 }−5n −14 = 0n

^{2 }−7n+2n −14 = 0n (n−7)+2(n −7) = 0

Solving for n,

(n −7)(n +2) = 0

Either n − 7 = 0 or n + 2 = 0

n = 7 or n = −2

Since, we know, n can neither be negative nor fractional.

Therefore, n = 7

From equation (i), we get

a = 6−2n

a = 6−2(7)

= 6−14

= −8

(ix)Given values are,first term, a = 3,

number of terms, n = 8

sum of n terms, S = 192

We know,

S

_{n}= n/2 [2a+(n -1)d]Substituting values,

192 = 8/2 [2×3+(8 -1)d]

192 = 4[6 +7d]

48 = 6+7d

42 = 7d

Solving for d, we get,

d = 6

(x)Given values are,l = 28,S = 144 and there are total of 9 terms.

Sum of n terms,

S

_{n}= n/2 (a + l)Substituting values, we get,

144 = 9/2(a+28)

(16)×(2) = a+28

32 = a+28

Calculating, we get,

a = 4

**Question 4. How many terms of the AP. 9, 17, 25 … must be taken to give a sum of 636?**

**Solution:**

Let us assume that there are n terms of the AP. 9, 17, 25 …

For this A.P.,

We know,

First term, a = 9

Common difference, d = a2−a1 = 17−9 = 8

Sum of n terms, is;

S

_{n}= n/2 [2a+(n -1)d]Substituting the values,

636 = n/2 [2×a+(8-1)×8]

636 = n/2 [18+(n-1)×8]

636 = n [9 +4n −4]

636 = n (4n +5)

4n

^{2}+5n −636 = 04n

^{2}+53n −48n −636 = 0Solving, we get,

n (4n + 53)−12 (4n + 53) = 0

(4n +53)(n −12) = 0

that is,

4n+53 = 0 or n−12 = 0

On solving,

n = (-53/4) or n = 12

We know,

n cannot be negative or fraction, therefore, n = 12 is the only plausible value.

**Question 5. The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.**

**Solution:**

Given:

first term, a = 5

last term, l = 45

Also,

Sum of the AP, Sn = 400

Sum of AP is equivalent to

S

_{n}= n/2 (a+l)Substituting the values,

400 = n/2(5+45)

400 = n/2(50)

Number of terms, n =16

Since, the last term of AP series is equivalent to

l = a+(n −1)d

45 = 5 +(16 −1)d

40 = 15d

Solving for d, we get,

Common difference, d = 40/15 = 8/3

**Question 6. The first and the last term of an AP are 17 and 350, respectively. If the common difference is 9, how many terms are there and what is their sum?**

**Solution:**

Given:

First term, a = 17

Last term, l = 350

Common difference, d = 9

The last term of the AP can be written as;

l = a+(n −1)d

Substituting the values, we get,

350 = 17+(n −1)9

333 = (n−1)9

Solving for n,

(n−1) = 37

n = 38

S

_{n}= n/2 (a+l)S

_{38}= 13/2 (17+350)= 19×367

= 6973

**Question 7. Find the sum of first 22 terms of an AP in which d = 7 and 22nd term is 149.**

**Solution:**

Given:

Common difference, d = 7

Also,

22nd term, a

_{22}= 149By the formula of nth term of an AP,

a

_{n}= a+(n−1)dSubstituting values, we get,

a

_{22}= a+(22−1)d149 = a+21×7

149 = a+147

a = 2 = First term

Sum of n terms,

S

_{n}= n/2(a+an)S

_{22}= 22/2 (2+149)= 11×151

= 1661

**Question 8. Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18, respectively.**

**Solution:**

Given:

Second term, a

_{2}= 14Third term, a

_{3 }= 18Also,

Common difference, d = a

_{3}−a_{2}= 18−14 = 4a

_{2}= a+d14 = a+4

Therefore,

a = 10 = First term

And,

Sum of n terms;

S

_{n}= n/2 [2a + (n – 1)d]Substituting values,

S

_{51}= 51/2 [2×10 (51-1) 4]= 51/2 [2+(20)×4]

= 51 × 220/2

= 51 × 110

= 5610

**Question 9. If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.**

**Solution:**

Given:

S

_{7}= 49S

_{17 }= 289Since, we know

S

_{n}= n/2 [2a + (n – 1)d]Substituting values, we get,

S

_{7}= 7/2 [2a +(n -1)d]S

_{7}= 7/2 [2a + (7 -1)d]49 = 7/2 [2a + 6d]

7 = (a+3d)

a + 3d = 7 ………………. (i)

Similarly,

S

_{17 }= 17/2 [2a+(17-1)d]Substituting values, we get,

289 = 17/2 (2a +16d)

17 = (a+8d)

a +8d = 17 ………………. (ii)

Solving (i) and (ii),

5d = 10

Solving for d, we get,

d = 2

Now, obtaining value for a, we get,

a+3(2) = 7

a+ 6 = 7

a = 1

Therefore,

S

_{n }= n/2[2a+(n-1)d]= n/2[2(1)+(n – 1)×2]

= n/2(2+2n-2)

= n/2(2n)

= n

^{2}

**Question 10. Show that a**_{1}, a_{2} …, a_{n}, … form an AP where an is defined as below

_{1}, a

_{2}…, a

_{n}, … form an AP where an is defined as below

**(i) a _{n }= 3+4n**

**(ii) a _{n} = 9−5n**

**Also**,** find the sum of the first 15 terms in each case.**

**Solutions:**

(i) a_{n}= 3+4nCalculating,

a

_{1 }= 3+4(1) = 7a

_{2}= 3+4(2) = 3+8 = 11a

_{3 }= 3+4(3) = 3+12 = 15a

_{4}= 3+4(4) = 3+16 = 19Now d =

a2 − a1 = 11−7 = 4

a3 − a2 = 15−11 = 4

a4 − a3 = 19−15 = 4

Hence, a

_{k}_{+ 1 }− a_{k}holds the same value between all pairs of successive terms. Therefore, this is an AP with common difference as 4 and first term as 7.Sum of nth term is;

S

_{n}= n/2[2a+(n -1)d]Substituting the value, we get,

S

_{15}= 15/2[2(7)+(15-1)×4]= 15/2[(14)+56]

= 15/2(70)

= 15×35

= 525

(ii) a_{n}= 9−5nCalculating, we get,

a

_{1}= 9−5×1 = 9−5 = 4a

_{2}= 9−5×2 = 9−10 = −1a

_{3}= 9−5×3 = 9−15 = −6a

_{4}= 9−5×4 = 9−20 = −11Common difference, d

a2 − a1 = −1−4 = −5

a3 − a2 = −6−(−1) = −5

a4 − a3 = −11−(−6) = −5

Hence, a

_{k + 1}− a_{k}holds the same value between all pairs of successive terms. Therefore, this is an AP with common difference as −5 and first term as 4.Sum of nth term is;

S

_{n}= n/2 [2a +(n-1)d]S

_{15}= 15/2[2(4) +(15 -1)(-5)]Substituting values,

= 15/2[8 +14(-5)]

= 15/2(8-70)

= 15/2(-62)

= 15(-31)

= -465