Magical A.P. Adventure

Discover the Arithmetic Patterns with Fun!

Check A.P. Sequences

Check whether these sequences are in A.P.:

a -3, a -5, a -7,...
1/2, 1/3, 1/4, 1/5,...
9, 13, 17, 21, 25,...
-1/3, 0, 1/3, 2/3,...
1,-1,1,-1,1,-1,...

An Arithmetic Progression (A.P.) has a constant difference between consecutive terms.

(i) a-3, a-5, a-7,...

Difference: (a-5)-(a-3) = -2, (a-7)-(a-5) = -2

Common difference (d) = -2 (constant)

✅ This is an A.P.

(ii) 1/2, 1/3, 1/4, 1/5,...

Differences: 1/3-1/2 = -1/6, 1/4-1/3 = -1/12

Differences are not equal (-1/6 ≠ -1/12)

❌ Not an A.P.

(iii) 9, 13, 17, 21, 25,...

Differences: 13-9=4, 17-13=4, 21-17=4

Common difference (d) = 4

✅ This is an A.P.

(iv) -1/3, 0, 1/3, 2/3,...

Differences: 0-(-1/3)=1/3, 1/3-0=1/3, 2/3-1/3=1/3

Common difference (d) = 1/3

✅ This is an A.P.

(v) 1,-1,1,-1,1,-1,...

Differences: -1-1=-2, 1-(-1)=2, -1-1=-2

Differences alternate between -2 and 2

❌ Not an A.P.

Find the A.P.

Given first term (a) and common difference (d), find the A.P.:

a = 5, d = 6
a = 7, d = -5
a = 3/4, d = 1/2

An A.P. is formed by repeatedly adding the common difference (d) to the first term (a).

(i) a = 5, d = 6

First term: 5

Second term: 5 + 6 = 11

Third term: 11 + 6 = 17

A.P.: 5, 11, 17, 23, 29,...

(ii) a = 7, d = -5

First term: 7

Second term: 7 + (-5) = 2

Third term: 2 + (-5) = -3

A.P.: 7, 2, -3, -8, -13,...

(iii) a = 3/4, d = 1/2

First term: 3/4

Second term: 3/4 + 1/2 = 5/4

Third term: 5/4 + 1/2 = 7/4

A.P.: 3/4, 5/4, 7/4, 9/4, 11/4,...

Find a and d from nth term

Find first term (a) and common difference (d) given nth term:

tₙ = 3n + 2
tₙ = 4 - 7n

General form of nth term: tₙ = a + (n-1)d

(i) tₙ = 3n + 2

Compare with general form: a + (n-1)d = 3n + 2

Expand: a + nd - d = 3n + 2

Rearrange: nd + (a - d) = 3n + 2

Compare coefficients: d = 3, a - d = 2 ⇒ a = 5

✅ a = 5, d = 3

(ii) tₙ = 4 - 7n

Compare with general form: a + (n-1)d = 4 - 7n

Expand: a + nd - d = -7n + 4

Rearrange: nd + (a - d) = -7n + 4

Compare coefficients: d = -7, a - d = 4 ⇒ a = -3

✅ a = -3, d = -7

Find the 19th term

Find the 19th term of A.P.: -11, -15, -19,...

Given A.P.: -11, -15, -19,...

First term (a) = -11

Common difference (d) = -15 - (-11) = -4

nth term formula: tₙ = a + (n-1)d

For 19th term (n=19): t₁₉ = -11 + (19-1)(-4)

Calculate: -11 + 18×(-4) = -11 - 72 = -83

✅ 19th term = -83

Which term is -54?

Which term of A.P. 16, 11, 6, 1,... is -54?

Given A.P.: 16, 11, 6, 1,...

First term (a) = 16

Common difference (d) = 11 - 16 = -5

nth term formula: tₙ = a + (n-1)d

Set tₙ = -54: 16 + (n-1)(-5) = -54

Solve: 16 -5n +5 = -54 ⇒ -5n +21 = -54

Continue: -5n = -75 ⇒ n = 15

✅ -54 is the 15th term

Find middle term(s)

Find middle term(s) of A.P.: 9, 15, 21, 27,…,183.

Given A.P.: 9, 15, 21, 27,…,183

First term (a) = 9, Common difference (d) = 6

Find number of terms (n): tₙ = a + (n-1)d = 183

9 + (n-1)×6 = 183 ⇒ (n-1)×6 = 174 ⇒ n-1 = 29 ⇒ n = 30

Since n=30 (even), there are two middle terms: 15th and 16th

15th term: t₁₅ = 9 + 14×6 = 9 + 84 = 93

16th term: t₁₆ = 9 + 15×6 = 9 + 90 = 99

✅ Middle terms: 93 and 99

Special A.P. Property

If 9×9th term = 15×15th term, show that 6×24th term = 0.

Given: 9×t₉ = 15×t₁₅

Express terms: tₙ = a + (n-1)d

9(a + 8d) = 15(a + 14d)

Expand: 9a + 72d = 15a + 210d

Rearrange: -6a = 138d ⇒ a = -23d

Now find 6×t₂₄ = 6(a + 23d) = 6(-23d + 23d) = 6×0 = 0

✅ Proved: 6×24th term = 0

Find k in A.P.

If 3 + k, 18 - k, 5k +1 are in A.P., find k.

For three terms in A.P., middle term is average of other two

So: 18 - k = [(3 + k) + (5k + 1)]/2

Multiply both sides by 2: 36 - 2k = 4 + 6k

Rearrange: 32 = 8k ⇒ k = 4

Verify: terms become 7, 14, 21 (common difference 7)

✅ k = 4

Find x, y, z in A.P.

Given x, 10, y, 24, z are in A.P., find x, y, z.

Given A.P.: x, 10, y, 24, z

Common difference (d) can be found from 10 to y to 24

10 + 2d = 24 ⇒ 2d = 14 ⇒ d = 7

Now find terms:

x = 10 - d = 10 - 7 = 3

y = 10 + d = 10 + 7 = 17

z = 24 + d = 24 + 7 = 31

✅ x = 3, y = 17, z = 31

Theater Seats Problem

Theater has 20 seats in front row, 30 rows total, each row has 2 more seats than previous. How many seats in last row?

This forms an A.P. where:

First term (a) = 20 seats

Common difference (d) = 2

Number of terms (n) = 30 (rows)

Last row seats = nth term = a + (n-1)d

Calculate: 20 + (30-1)×2 = 20 + 58 = 78

✅ Last row has 78 seats

Three Consecutive Terms

Sum of 3 consecutive A.P. terms is 27, product is 288. Find the terms.

Let terms be: a - d, a, a + d

Sum: (a-d) + a + (a+d) = 3a = 27 ⇒ a = 9

Product: (9-d)×9×(9+d) = 288

Simplify: 9(81 - d²) = 288 ⇒ 729 - 9d² = 288

Rearrange: 9d² = 441 ⇒ d² = 49 ⇒ d = ±7

Possible terms:

If d=7: 2, 9, 16

If d=-7: 16, 9, 2

✅ Terms are 2, 9, 16

Ratio of Terms

Ratio of 6th to 8th term is 7:9. Find ratio of 9th to 13th term.

Given: t₆/t₈ = 7/9

Express terms: [a + 5d]/[a + 7d] = 7/9

Cross multiply: 9a + 45d = 7a + 49d

Simplify: 2a = 4d ⇒ a = 2d

Now find t₉/t₁₃ = [a + 8d]/[a + 12d]

Substitute a=2d: [2d + 8d]/[2d + 12d] = 10d/14d = 5/7

✅ Ratio is 5:7

Temperature in Ooty

Ooty temperatures Monday-Friday in A.P. Sum Mon-Wed is 0°C, Wed-Fri is 18°C. Find each day's temperature.

Let temperatures be: a-2d, a-d, a, a+d, a+2d

Mon-Wed sum: (a-2d) + (a-d) + a = 3a - 3d = 0 ⇒ a = d

Wed-Fri sum: a + (a+d) + (a+2d) = 3a + 3d = 18

Substitute a=d: 3d + 3d = 6d = 18 ⇒ d = 3

Thus a = d = 3

Calculate temperatures:

Mon: a-2d = 3-6 = -3°C

Tue: a-d = 3-3 = 0°C

Wed: a = 3°C

Thu: a+d = 6°C

Fri: a+2d = 9°C

✅ Temps: -3°, 0°, 3°, 6°, 9°

Priya's Savings Problem

Priya earns ₹15,000 first month, salary increases by ₹1500/year. Expenses ₹13,000 first month, increase by ₹900/year. When will she save ₹20,000/month?

Salary A.P.: a₁ = 15000, d₁ = 1500/month after 1 year = 125/month

Expenses A.P.: a₂ = 13000, d₂ = 900/year = 75/month

Monthly savings: (15000 + 125(n-1)) - (13000 + 75(n-1))

Simplify: 2000 + 50(n-1) = 1950 + 50n

Set savings = 20000: 1950 + 50n = 20000

Solve: 50n = 18050 ⇒ n = 361 months = 30 years 1 month

✅ After 30 years and 1 month

$Math Wizard$