Find the common difference of am AP: $- 225, - 425, - 625, - 825, . . . . .$

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Solution

Given sequence is,
$- 225, - 425, - 625, - 825, . . . \dots$
first term of this A.P is $a_{1} = - 225$
second term of this A.P is $a_{2} = - 425$
third term of this A.P is $a_{3} = - 625$
fourth term of this A.P is $a_{4} = - 825$
the condition for an sequence to be an A.P is their must be a common difference $(i . e ., d = a_{n + 1} - a_{n})$
putting n=1 in above equation
$d = a_{2} - a_{1} = (- 425) - (- 225) = - 200$
putting n=2 in above equation
$d = a_{3} - a_{2} = (- 625) - (- 425) = - 200$
putting n=3 in above equation
$d = a_{4} - a_{3} = (- 825) - (- 625) = - 200$

as we can see we get a common difference $d = - 200$ for this sequence
hence this sequence forms an A.P
with first term $a_{1} = - 225$ and
common difference $d = - 200$

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Q. Which of the following sequences are arithmetic progressions. For those which are arithmetic progressions, find out the common difference.

(i) 3, 6, 12, 24, ...

(ii) 0, −4, −8, −12, ...

(iii)

\frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, . . .

(iv) 12, 2, −8, −18, ...

(v) 3, 3, 3, 3, ...

(vi) p, p + 90, p + 180 p + 270, ... where p = (999)⁹⁹⁹

(vii) 1.0, 1.7, 2.4, 3.1, ...

(viii) −225, −425, −625, −825, ...

(ix) 10, 10 + 2⁵, 10 + 2⁶, 10 + 2⁷,...

(x) a + b, (a + 1) + b, (a + 1) + (b + 1), (a + 2) + (b + 1), (a + 2) + (b + 2), ...

(xi) 1², 3², 5², 7², ...

(xii) 1², 5², 7², 73, ...

Q. Find the common difference of an AP

Find out which of the following sequences are arithmetic progressions. For those which ar3e arithmetic progressions, find out the common difference.
(i) $3, 6, 12, 24 \dots$
(ii) $0, - 4, - 8, - 12 \dots$
(iii) $\frac{1}{2}, \frac{1}{4}, \frac{1}{6} . \frac{1}{8}, \dots$
(iv) $12, 2, - 8, - 18, \dots$
(v) $3, 3, 33, 3, \dots$
(vi) $p, p + 90, p + 180, p + 270, \dots$ where $p = (999)^{999}$
(vii) $1.0, 1.7, 2.4, 3.1, \dots$
(viii) $- 225, - 425, - 625, - 825, \dots$
(ix) $10 + 10 + 2^{5}, 10 + 2^{6}, 10 + 2^{7}, \dots$
(x) $a + b, (a + 1) + b, (a + 1) + (b + 1), (a + 2) + (b + 1), (a + 2) + (b + 2), \dots$
(xi) $1^{2}, 3^{2}, 5^{2}, 7^{2}, \dots$
(xii) $1^{2}, 5^{2}, 7^{2}, 7^{3}, \dots$

Q. Find the common difference of an AP:

1^{2}, 5^{2}, 7^{2}, 73, . . . .

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Find the common difference of am AP: −225,−425,−625,−825,.....

Find the common difference of am AP: $- 225, - 425, - 625, - 825, . . . . .$