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Last updated date: 02nd Dec 2023
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# If $a,b,c,d$are any four consecutive coefficients of any expanded binomial, then $\frac{{a + b}}{a},\frac{{b + c}}{b},\frac{{c + d}}{c}$are inA. A.P.B. G.P.C. H.P.D. None of the above

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Hint
The process of expanding and writing terms that are equal to the natural number exponent of the sum or difference of two terms is known as binomial expansion. The exponents of both terms added together in the general term equal n, and the coefficient values are obtained from the Pascal's triangle or by applying the combinations formula. When a bracket is expanded, each term is multiplied by the expression outside of the bracket.
A binomial is a two-term algebraic expression that includes a constant, exponents, a variable, and a coefficient. We must multiply out the brackets in order to expand and simplify an expression, and we must then collect like words in order to simplify the resulting expression.
Formula used:
The arithmetic progression is $a,b,c$
$(b - a) = (c - b)$
The geometric progression is $a,b,c$
${b^2} = ac$.

Complete step-by-step solution
The expansion of the series be ${(1 + x)^n}$
$a,b,c,d$be the $(r + 1)$th , $(r + 2)$th, $(r + 3)$th and $(r + 4)$th coefficients.
So, $a{ = ^n}{C_r},b{ = ^n}{C_{r + 1}},c{ = ^n}{C_{r + 2}},d{ = ^n}{C_{r + 3}}$
Here, $\frac{a}{{a + b}} = \frac{{^n{C_r}}}{{^n{C_r}{ + ^n}{C_{r + 1}}}}$
$\frac{{^n{C_r}}}{{^{n + 1}{C_{r + 1}}}} = \frac{{n!}}{{r!(n - r)!}} \times \frac{{(r + 1)!(n - r)!}}{{(n + 1)!}}$
$= > \frac{{r + 1}}{{n + 1}}$
Likely, $\frac{b}{{b + c}} = \frac{{(r + 1) + 1}}{{n + 1}} = \frac{{r + 2}}{{n + 1}}$
$\frac{c}{{c + d}} = \frac{{(r + 2) + 1}}{{n + 1}} = \frac{{r + 3}}{{n + 1}}$
The series in A.P are
$= > \frac{a}{{a + b}},\frac{b}{{b + c}},\frac{c}{{c + d}}$
The series in H.P are
$= > \frac{{a + b}}{a},\frac{{b + c}}{b},\frac{{c + d}}{c}$
Therefore, the correct option is C.

Note
The number of ways to choose unordered results from potential outcomes is known as the binomial coefficient, commonly referred to as a combination or combinatorial number. When used to represent a binomial coefficient, the symbols and are sometimes read as " select ".
Each row is constrained on both sides by 1 and each coefficient is derived by adding two coefficients from the preceding row, one on the immediate left and one on the immediate right.