Assume that a1=-5, ∑i=1^10 ai=20, and ∑i=1^10 bi=7 . Calculate: (a)∑i=1^10(4 ai+3) (b) ∑i=2^

step 1 Okay, so in this question, we're asked to calculate three different sums, and we're given some i

Assume that a1=-5, ∑i=1^10 ai=20, and ∑i=1^10 bi=7 ....

Assume that a_{1}=-5, \sum_{i=1}^{10} a_{i}=20, and \sum_{i=1}^{10} b_{i}=7 . Calculate:
$$
\text{(a)}\sum_{i=1}^{10}\left(4 a_{i}+3\right) \quad \text { (b) } \sum_{i=2}^{10} a_{i} \quad \text { (c) } \sum_{i=1}^{10}\left(2 a_{i}-3 b_{i}\right)
$$

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Key Concepts

Linearity of Summation

This concept refers to the fact that the summation operator is linear, meaning that the sum of a linear combination of sequences can be separated into the sum of individual sequences multiplied by their respective constants. In abstract terms, for any sequences {x_i} and {y_i} and constants c and d, the property is expressed as ?(c*x_i + d*y_i) = c*?x_i + d*?y_i.

Summation of Constants

This concept involves recognizing that when summing a constant over a set of indices, the result is simply the constant multiplied by the number of terms. More specifically, if you sum a constant k over n terms, the total is k*n, which is crucial when handling terms like adding the same number repeatedly within a summation.

Adjustment of Summation Limits

This concept addresses how changes in the limits of summation affect the sum. Sometimes, the sum over a subset of indices (like i = 2 to n) must be expressed in terms of the sum over a larger set (like i = 1 to n) by subtracting the missing terms. This adjustment is important for correctly computing sums when the range of indices is shifted.

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