Let \({a_k}\) and \({b_k}\) , \(𝑘∈N\) , be two G.P.s with common ratios \(r_1\) and \(r_2\) respectively such that \(a_1= b_1=4\) and \(r_1<r_2\) . Let \(c_k=a_k+b_k\) , \(k∈N\) . If \(c_2=5\) and \(c_3=\frac {13}{4}\) , then \( \displaystyle\sum_{k=1}^{∞ }c_k−(12a_6+8b_4)\) is equal to _____ .

Given that ck = ak + bk and a1 = b1 = 4 also a2 = 4r1 a3 = 4r1² b2 = 4r2 b3 = 4r2² Now c2 = a2 + b2 = 5 and c3 = a3 + b3 = 13/4 ⇒ r1 + r2 = 5/4 and r1² + r2² = 13/16 Hence r1r2 = 3/8 which gives r1 = 1/2 and r2 = 3/4 ∑ k=1 ∞ ck - (12a₆ + 8b₄) = 4 / (1 - r1) + 4 / (1 - r2) - (48 / 32 + 27 / 2) = 24 - 15 = 9

Let ak and bk,𝑘∈ℕ, be two G.P.s with common ratios r1 and r2 respectively such that a1=b1=4 and r1<r2.Let ck=ak+bk,k∈ℕ.If c2=5 and c3=13/4 then ∑k=1∞ ck−(12a6+8b4) is equal to_

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Concepts Used:

Sequence and Series

Sequence: Sequence and Series is one of the most important concepts in Arithmetic. A sequence refers to the collection of elements that can be repeated in any sort.

Eg: a₁,a₂,a₃, a₄…….

Series: A series can be referred to as the sum of all the elements available in the sequence. One of the most common examples of a sequence and series would be Arithmetic Progression.

Eg: If a₁,a₂,a₃, a₄……. etc is considered to be a sequence, then the sum of terms in the sequence a₁+a₂+a₃+ a₄……. are considered to be a series.

Types of Sequence and Series:

Arithmetic Sequences

A sequence in which every term is created by adding or subtracting a definite number to the preceding number is an arithmetic sequence.

Geometric Sequences

A sequence in which every term is obtained by multiplying or dividing a definite number with the preceding number is known as a geometric sequence.

Harmonic Sequences

A series of numbers is said to be in harmonic sequence if the reciprocals of all the elements of the sequence form an arithmetic sequence.

Fibonacci Numbers

Fibonacci numbers form an interesting sequence of numbers in which each element is obtained by adding two preceding elements and the sequence starts with 0 and 1. Sequence is defined as, F₀ = 0 and F₁ = 1 and F_n = F_n-1 + F_n-2

Let \({a_k}\) and \({b_k}\), \(𝑘∈N\), be two G.P.s with common ratios \(r_1\) and \(r_2\) respectively such that \(a_1= b_1=4\) and \(r_1<r_2\). Let \(c_k=a_k+b_k\), \(k∈N\). If \(c_2=5\) and \(c_3=\frac {13}{4}\), then \( \displaystyle\sum_{k=1}^{∞ }c_k−(12a_6+8b_4)\) is equal to _____ .

Correct Answer: 9

Solution and Explanation

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