(a)

#### To determine

**To find:** The equation for
Sn.

#### Answer

The equation for
Sn is
D(1−cn)1−c.

#### Explanation

**Given:**

The concentration of insulin in a patient’s system is
De−at.

**Result used:**

The sum of the finite geometric series
(∑k=1nark−1 (or) a+ar+ar2+⋯+arn−1) is
a(1−rn)1−r, where *a* is the first term and *r* is the common ratio of the series.

**Calculation:**

Let
Sn be represents the amount of total spending done by *n* recipients.

Initially, the government spent *D* dollars to the economy by giving it to a recipient. That is,
S1=D.

The recipient who receives the *D* dollars, will spend *Dc* dollars and the amount of total spending after done by 1 recipients is
S2=D+Dc.

The recipient who receives the *Dc* dollars, will spend
Dc⋅c=Dc2 dollars and the amount of total spending done by 2 recipients is
S3=D+Dc+Dc2 .

The recipient who receives the
Dc2 dollars, will spend
Dc2⋅c=Dc3 dollars and the amount of total spending done by 3 recipients is
S4=D+Dc+Dc2+Dc3.

Proceed in the similar way, the *n* recipient will get
Dcn−1 dollars, will spend
Dcn−1⋅c=Dcn.

The amount of total spending done by *n* recipients computed as follows,

Sn=D+Dc+Dc2+Dc3+⋯+Dcn=∑k=0nDck

This is geometric series with the first term of the series is
a=D and the common ratio is
r=c.

Use the Result stated above, the sum of the series
Sn=D(1−cn)1−c.

(b)

#### To determine

**To show:**
limn→∞Sn=kD, where
k=1s.

#### Answer

The multiplier is
k=5.

#### Explanation

**Given:**

c+s=1 and s=1k

**Proof:**

From part (a),
Sn=D(1−cn)1−c with
0<c<1.

Take limit on both sides,

limn→∞Sn=limn→∞D(1−cn)1−c=D1−c⋅limn→∞(1−cn)=D1−c(limn→∞1−limn→∞cn)=D1−c(1−limn→∞cn)

If
0<c<1, then
limn→∞cn=0.

limn→∞Sn=D1−c(1−0)=D1−c(1)=D1−c

Since
c+s=1 and s=1k,
s=1−c and k=1s.

limn→∞Sn=D1−c=Ds=1s⋅D=kD

Hence the required proof is obtained.

Now find the multiplier *k.*

**Calculation:**

The marginal propensity to consume is
c=80%. That is,
c=0.8.

Since
s+c=1,
s=1−c.

Substitute 0.8 for *c*,

s=1−0.8=0.2

Thus, the value of
s=0.2.

The multiplier *k* is computed as follows.

k=1s=10.2=102=5

Therefore, the multiplier is
k=5.