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(1) Addition/Subtraction of constant number to each term of an A.P. also results an A.P.
e.g. suppose a1, a2, a3, …, an are in A.P.
then a1 + k, a2, k, a3 + k ……, an + k and
a1 –k, a2 – k, a3 – k, ……, an–k will also be in A.P.
where k ? R
(2) Multiplication/Division by a constant number to each term of an A.P. also results an A.P.
e.g. suppose a1, a2, a3, …, an are in A.P.
then ka1, ka2, ka3 ……, kan and
a1/k, a2/k, a3/k will also be in A.P.
where k ? R and k ≠ 0
(3) Addition/Subtraction of two A.P.’s also results an A.P.
e.g. suppose a1, a2, a3, …, an and
b1, b2, b3, …, bn, are in A.P.
then a1+b1, a2+b2, a3+b3, ……, an+bn.
a1–b1, a2–b2, a3–b3, ……, an–bn will also be in A.P.
(4) Reversing the order of an A.P. also results an A.P.
e.g. Suppose a1, a2, a3, ……, an are in A.P.
then an, an–1, ……, a3, a2, a1 will also be in A.P.
Note:
Students will find various problems related to this chapter, where they will need to assume (as unknown) three, four or five numbers are in A.P. Normally it is found that if you assume such numbers in following way then it becomes easy to solve the problems. Three numbers are in A.P.: α–ß, α, α+ß (here first term is α–ß and c.d. is ß). Four numbers are in A.P.: α–3ß, α–ß, α + ß, α+3ß (here first term is α-3ß and c.d. is 2bß. Five convenient numbers in A.P. are α–2 ß, α–ß, α–ß, α, α + ß, α + 2 ß.
Basically sum of the above sets of variables eliminate one variable which make easy to solve the problems.
Note:
1. If nth term of an series is tn = A n + B, then
tn = tn–1 = A n + B – A(n–1)–B
= A = constant
i.e. the series is in A.P.
2. If tn = A n2 + B n + c, then
tn – tn–1 = A(n2) + Bn + C – (A(n–1)2 + B(n–1) + c)
= A (2n + 1) + B = ∪
(n) (say)
Un – Un–1 = 2A
which is constant
i.e. if the difference of difference of the terms of a series is constant, then the nth term is quadratic in n.
3. Similarly, if the difference the terms of a series is constant then the nth term of the series is a cubic in n i.e.
tn = An3 + Bn2 + Cn + D
and so on.
Summary of Important Notes:
• If a fixed number is added to (or subtracted from) each term of a given A.P., then the resulting sequence is also an A.P. with the same common difference as that of the given A.P.
• If each term of an A.P. is multiplied by a fixed number (say k) (or divided by a non-zero fixed number), the resulting sequence is also an A.P. with the common difference multiplied by k.
• If a1, a2, a3, …… and b1, b2, b3, …… are two A.P.’s with common differences d and d’ respectively then a1 + b1, a2 + b2, + a3 + b3, … is also an A.P. with common difference d + d’.
• If we have to take three terms in an A.P., it is convenient to take them as a – d, a, a + d. In general, we take a – rd, a – (r – 1)d, …., a – d, a, a + rd in case we have to take (2r + 1) terms in an A.P.
• If we have to take four terms, we take a – 3d, a – d, a + d, a + 3d. In general, we take a –(2r–1)d, a –(2r–3)d,…, a–d, a+d, …, a + (2r–1)d, in case we have to take 2r terms in an A.P.
• If a1, a2, a3, ……, an are in A.P., then a1 + an = a2 + an–1 = a3 + an–2 = …… and so on.
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